Found my answer today and learned a lot more things. The bike park I frequent is an elaborate spider nest of dozens of other bike parks all (poorly) weaved together. That's why I couldn't find the link from my side to across the river, the pedestrian access points are below most of the bridges that cross it.
You need some really keen eyes to see a sign that says "Yonghe Bridge" and understand it's labeled because you can access it. I actually passed it at first. On the way back from that route, I took another sign that was labeled "Yonghe Bridge" that ended up being a completely different bridge.
Alright, so here's my (early) opinions to the side I frequent (New Taipei) as compared to the eastern side (Taipei City). Personally I think the former is much more open and less crowded. There's not as many confusing twists and turns, and still plenty scenic viewing areas as well as exercise equipment throughout.
However, the other side has its advantages. I noticed more exercise equipment, and a little more variety. The scenic areas were a little more diverse as well. And most importantly it had more vending machines throughout.
That said, I did find it (Taipei City areas) to be very confusing since most of the paths I took would break off into multiple sections whereas I was accustomed to one big loop. And it was far more congested. [Oh, on an aside there is a very convenient bike shop right across the bridge in Taipei City right as you exit. I could clearly see many miscellaneous parts, and some cold drinks. And yes, I know some of the pedestrian bridges can be accessed from the road if you know what you're looking for. I found that out much later]
Next time I might just go up the bridge for a break and enjoy the view, then continue back down to my area. This picture doesn't do it justice, I was battling the incoming rain.