Does felszáll also work?
No. "Felszáll" means "get up" only when used with means of transportation. Felszáll a buszra, a vonatra, a villamosra, stb.
"Felmegy", however, should be accepted.
"Felszáll" can also mean take off (e.g. an airplane) - of course not in this sentence, unless this car can also fly ;)
Perhaps it belongs to a certain k**********n t*****r.
Am I dating myself if I say, "Chitty-chitty Bang-bang"?
Yes, but it's such a wonderful reference that I can only admire it.
"Gets up on the sidewalk" is about as ghastly as it gets when it comes to speaking English. Nobody speaks like that. Parks up on the sidewalk, (pavement please for those of us who are English!), is fine.