About the parallax: It gets a little funky when I am climbing up hills at the end of the first main area. If I'm standing in one of those dips and move left to jump out of it and onto the ledge, it seems as though the background anticipates my movement and moves just slightly before I do. Same for if I'm changing directions as I walk. I like how the different background elements scroll in different ways as I climb the giant hill at the end of the second main area. The clouds should get some parallax!
I'm not sure if it was the engine or my computer, but whenever you changed rooms there would be a significant time lag that sometimes hung for up to a minute.
Seconded. This stood out to me as the worst aspect of the demo, and sometimes made me wonder if the game had frozen.
A small glitch I noticed immediately after I got the gun: one frame of the gun blast animation stayed onscreen once when I shot down while jumping.
I think the mayor's office should include some office props, like a desk, plaque, etc. The building interiors all look the same. Jables should probably have a "hurt" animation before you finish the game, too, as should the enemies, so you know your shots have done some damage.
I was disappointed that I couldn't talk to the keyboard cat. The squid in the cave by the spike-pit didn't say anything to me, either.
Love the bits of humor, like the squid's useless advice about jumping, and Jables muttering about how he couldn't buy anything.
Cool tunes, especially the cave theme. I got a little bit tired of the main village exploration theme.
Ho-ly crap, the hills have eyes! THAT freaked me out there.

Nice. Er...will that become important to the game? Because that is very, very memorable.
You guys are off to a promising start with this. Good luck!