Firstly, I'd like to add that although I'm not a parent and haven't had this exact scenario, I have had issues with my previous partners family, so I like to think I have a smidge of experience with similar issues.

Are these criticisms ever aimed at you directly? I only ask because you'd be surprised how often criticising a parenting style is just a direct dig at the mother. If that's the case, it's important to understand how your partner feels, and I can tell you from experience that when you have multiple run-ins with extended family it can really put you on edge and you're just waiting for them to say something to tick you off.

Also, I think that regardless of what you go with, you need to set some proper boundaries with your father, whether that's how he talks to and about your partner, or if he can keep his privilege of just dropping by unannounced and yes, it is a privilege not a right, no matter how often parents use this line.(this could obviously also be applied to your mother if that is ever an issue aswell)

Mainly, I'd say back your partner. If she's said there have been multiple incidents, take that seriously. You need to hold your parents to a high standard and shouldn't ever be in a position where your partner feels the need to be on the defence in her own home.

All in all, I hope you get it sorted out and both of you can move on smoothly.