She feels I insulted her parents...

My girlfriend and I recently had an argument about something her parents said, and she has been non-communicative since then. She says I insulted her parents, which is not true. She can’t stand anyone disagreeing with them, and that is not the same as someone insulting them. She has always been very touchy about them, and I have been aware of this, but it hasn’t been such a big problem until now. My worry is that this will affect our relationship going forward, because I will constantly have to censor myself if I ever say something involving them. Is this a sign that this relationship may fail?
If you can’t be honest about how you feel and are always going to be afraid of being misinterpreted, you will both have to work on your communication skills. Being touchy about something isn’t a problem, provided it doesn’t spill over into affecting other aspects of your life, which is what appears to be happening here. I suggest you tell your girlfriend why her sensitivity may be justified, but why it is also preventing you from being honest about how you feel. Agreeing to disagree about something is a mark of maturity, and a critical aspect of any relationship between adults. If you can’t get to that place, you will both have to think about what this means for your future together.

I have a problem with something my girlfriend recently said, because it implied that she wasn’t happy with how supportive I have been. She made it seem as if she was doing all the work in this relationship. She won’t elaborate so I don’t know what to do. How can I fix this?
You can evaluate every aspect of what you say or do and be more cognisant of what has an adverse effect on her. Having said that, she needs to understand that being open is always the preferred approach to any issue.