You don't need parental consent to go to a doctor unless she's young enough that she shouldn't be having sex to begin with. There are barrier methods besides condoms you might look into. I'd highly recommend she at least try the pill before considering the shot. It has some much more serious side-effects including depression, suicidal thoughts, and other really awful drug reactions, and they will last for the full time the drug is supposed to be effective. And actually, the most effective form of body birth control is one YOU use:
http://techcitement.com/culture/the-best-birth-control-in-the-world-is-for-men/?fb_ref=.T3OMH1fzl5M.like&fb_source=home_oneline
It is safe, drug-free, limited side-effects, highly effective, and completely reversible.
All that Now-You-Know out of the way...
Unless your girlfriend's mother is likely to be abusive or break with her daughter over this, she really ought to be honest with her. You're facilitating her in betraying her mother's trust and damaging their relationship even farther with increasingly important decisions from which she is being excluded with by lies. There are circumstances that justify this. But I think you need to ask yourself, and then your girlfriend, if these are those circumstances. Does she fear her mother will do something drastic? Or is she simply too ashamed to tell her? If it's the latter, you might have some reevaluating to do about your relationship on various levels.