Basically the argument for Peter being Jewish comes down to "Stan Lee is Jewish, Peter Parker is his best known character, therefore Peter is Jewish." There isn't really any basis for it. They explicitly avoided any mentions of religious stuff for years, and when they started doing them it was all Christmas trees and stuff.
His relatives' names are Parker, Reilly, and Fitzpatrick, which don't really scream Jewish to me. Someone Who Actually Knows Things might have mixed Ben Parker up with Ben Grimm.
Bendis kind of played around with that in Ultimate Spider-Man, having Peter using Yiddish expressions all the time, yet being explicitly not Jewish, in contrast to Kitty Pryde.
Stan & Co. (and I think especially Jack Kirby) went out of their way to include varied characters with a minimum of racial stereotyping, but I think were under some pressure to be subtle about it.