I don't think it really matters if you make it at a smaller resolution or in full hd. Scaling a 320x240 game up on an hd monitor is going to look a bit shitty mind!
But probably not as bad if you were to scale it up from say 1024x768.
I don't understand why 800px is needed for parallax scrolling or why games in hd look static... you can add parallax scrolling & life to either. If you were to make the game in HD then you would just need to make the scenes/objects/characters etc bigger than you would @ 800px or whatever.
Typical resolutions these days is:
640x480 (480p) - pretty low & looks blurry on hd monitors... hell doesn't look great on my laptop screen either...
1280x720 (720p) -- decent middle ground
1920x1080 (1080p) -- standard hd
Anyway I suppose it all depends on the art style of your game. Pixel based games tend to use lower resolutions whereas vector/digital tends to use middle to high resolutions.
When you export/build your game you can actually set a custom size to export it at... so you can make the game at 1920x1080 if you liked & then set a custom resolution for certain platform builds, as well as webP image conversion to reduce image size/format for mobile platforms to save space/memory.