No, you cannot do this with Google app engine without significant coding to manage everything.
Parse makes all of this insanely easy, so I would suggest if you're not a backend developer I would stick with Parse and worry about what happens if you get too many users later.
In the unlikely event that your app is wildly successful and you have a plan to monetize it, $200 per month shouldn't be a problem. I don't know enough about India though so maybe that's a burden. In that case, with a nicely polished app that showcases Parse's capabilities, I would be surprised if you couldn't get in touch with Parse directly and work something out.
You could also try Googling for "parse alternatives" and see what other services offer.