What does free parking in the cities look like, and where is one likely to find it?
Is it OK to parallel park where there is a single white line?
What happens if you get a parking ticket (not fine) on your windscreen, go off to pay it and while you are gone, get another one?
How severe are the fines for parking improperly, and could you potentially also get points on your licence?
Just park wherever itās convenient and turn on the hazard lights. Donāt worry about parking on the sidewalk or something - the pedestrians will just move around your car 
On a more serious note: I donāt think youāll really find free parking in a city like Taipei, for example, except when visiting a mall / store and fulfilling their conditions for free parking (minimum purchase, membership, ā¦).
Turning on your hazard lights, to do anything you want, is indeed in the last class for the best students, just before one goes for their driving test.
Is it free to park EVs in all the cities or just some ?
My wife actually got a parking ticket for parking on a red line with hazards on and going into a store. Didnāt think it was possible, should frame the ticket.
There is no universal Taiwan parking standard because every city or district makes their own rules but generally
Red Line - no parking ever
Yellow line - parking during certain periods like weekends or at night depending on location
White line - free parking but may need to be outside the line from the road
No line - free parking
Read the signs on the sign posts or spray painted on the road.
Holidays are free.
You have to pay for parking that is placed on your windshield even if you are on your way just paying for the most recent one.
Short-term like for an hour or overnight free parking in Taipei for example itās very difficult to find. Any free parking is filled up with people who park their cars for days or weeks or months at a time and then it only moves it and swap it with a different car so they can get their spot back when they return.
Iāve paid parking fines from $1,500 up to 3,000.
One time I parked on a pedestrian crossing while in 7-Eleven buying a bottle of water and they ticketed me and I never saw anyone close to the car. It was like wham bam thank you sir.
There are free parking spots in Taipei where I see the same cars parked for years. One time I saw the owner came just to idle the car for a while but didnāt move 
Itās practically impossible to park for free in Taipei & most of New Taipei during weekdays, but itās quite OK to find a free parking spot in the evening/weekend if youāre willing to circle around the block and walk a little (like parking 1km from your destination)
What I usually see in places like Danshui or Guanghua is a long line of Taiwanese queueing for the parking lot, but if you drive further you can find free/much cheaper parking spot in some side alley, but of course then you need to walk couple hundred meters which I guess is why locals still prefer to queue 
You need to drive around and explore areas away from everything.
I have not used free parking recently, but previously in Taipei there was free parking near the river in Nangang Tech park. Also, behind the Songshan airport.
Next to most rivers also has paid parking that is cheaper than on normal roads. Many people park there during the weekends. Be careful about typhoon warnings as your car may be towed or get flooded.
The meter folks are often very kind. Many times they have seen me in the car, knocked politely, and asked if I was leaving, staying a short time, or longer. If you say just a minute or two, they usually dont charge and tell you they will complete one round before being back.
I thought it quite nice.
And where was this? Taipei? In my experience you need to actively wave them away not to get a ticket.
Danshui seemed to have a fair number of abandoned cars, some pretty nice but older models, particularly up the hill a bit between Tamsui MRT and Hongshulin.
Sanxia, Tucheng, and Banqiao.
All jokes aside, the rules are always clear and standard in Taiwan. it is the enforcement that differs. And realistically, that is an important distinction.
You can often get fined parking in illegal places. You also often dont. These norms exist together. Better to just park legally. Especially with the citizen police about.
as for free parking, that is the goal in every big city. the best we get in taipei is 30/day. free exists I hear, but never have experienced it first hand. the ones that are free are normally crammed right tight. The few times we see exceptional parking skills in Taiwan : Taipei free parking spots on busy roads. I mean, millimetres ććć
Is there a joke thing going about parking cars?
Anything I said about parking cars was not a joke so which ones were just talking b******* jokes?
Pretty much this.
Only thing isā¦ remember you cannot park on a yellow line unless itās indicated somewhere. Yellow lines are for dropping or picking up people or stuff, not for parking.
Iāve parked for free on both legal and illegal spots in Taipei.Actually Iāve been pretty lucky many times, but many others got my car towed away. Theyāve been REALLY busy in the past monthsā¦
You can find it at some of temples, especially on the mountains outside the city center.
There was a ton of bullshit above, just read the replies. Though if they were being serious, I am seriously speechless 
When you say outside the line from the road, do you mean already onto the sidewalk or off the asphalt?
Are there times when there is a white line or no line, and yet the sign posts say that there are parking restrictions, such as for residents in nearby apartments only?
Iāve seen some places where there are white lines in front of shops, but the shops have ramps in front of them as if a vehicle could back into the shop perhaps to facilitate loading. Is it possible to park there for free, or could I get my car keyed by the owner of the shop?
Previously, I used to find free parking in supermarket lots but science parks are a good idea to try.
What about free parking near transport hubs to encourage people to take public transport?
Also residential areas during working hours, especially cul-de-sacs and low traffic areas.
I do not know. I never figured out how you know whether an area is paid parking or not. There are signs but it gets confusing. I even asked a cop at one point about a street I was on and he had no idea or was unwilling to help. In general, I do not take risks to park in front of peopleās houses if there is no side-walk, because most people think the space in front of their house belongs to them. I just do not want the trouble. My car has been scratched up before.
So in general, my approach was to ask people I know or just āgive it a tryā and go check on my car later. The worst that can happen is that you get a full day of paid parking. Paid parking is usually not too expensive if you park far out of the way. The positive is that if you invest time and effort, you can probably find ways to significantly reduce parking costs (not free though). I used to have a fold-up bike in the back of my car for this purpose.
The bad thing is that you will waste a lot of time in very hot weather (in summer) walking or biking back from your car. And sometimes you cannot find parking quickly and you will spend an hour or two.
If you can get free parking at work that will help, as you only need to find free parking after hours near the house. Still, I would have to pay to park near home on Saturdays.
For now, I leave my car permanently at the office and I rarely drive it.
betelnut?
There is lots of free parking up the mountains around taipei. E.g. up in beitou, tucheng, neihu, etc.
This is most indicated by cars parked there for months without moving.
Of course it is a hike to get up And down.
But donāt park there without checking regularly.
Sometimes the authorities also change a white line to a red lines, to get rid of the broken down cars.
For almost free parking (30 TWD/day), i see it along most rivers. (E.g. danshui river, in shilin waishuangxi, etc.)
You just have to go to seven every week to pay your parking fees. you donāt need the actual ticket from the car, just use ibon and key in the number plate.
Thatās why all those long parked cars have huge bundles of parking tickets.