Screen part that holds one side of screen in

That looks like the foam spline to me. There are many different sizes so you'll have to measure yours to confirm which one to go with. You can see our complete selection here.

Thanks for clarifying. This is a latch of some kind, but it's not like any latch I've seen before. Could there be more to it, maybe on the other side of the door? Do you also know who made the door to begin with?

I wasn't able to find a match for this, either. The only Pella screen latch we have is the 70-218 but this is obviously a different shape compared to your original. Without seeing it out of the frame I'm not sure what to look for. I think your best bet at this point would be to contact Pella directly, or maybe a store near you who deals in Pella hardware. This might be ab older part that's not available to us anymore.