Good advice
Parallax is an overrated demand people want. And yes, if you're just learning, it will only result in extra headaches.
Lots of really great PCE games don't even use parallax, so remember that when people hassle you about no parallax.
and then point them to Ninja Gaiden for an example of shit parallax, lol.
It looks good though. I am looking forward to you getting frustrated while implementing the red gargoyle thingy that attacks you in level 1. \o/