Good communication can't solve sexism. But it can help a husband and wife support each other and work on the same team.
"Hey honey. I'm really sick of writing these cards. I know you don't care, but if I don't write them your mum gets snippy with me for not sending her a card"
"Well just don't send them and let her be snippy?"
"I hate that, I feel judged by her for doing that. And I feel frustrated because she's your mom and this is somehow my problem. I'd appreciate some support with this."
"I'm sorry you hear that way. I can tell you're really frustrated about this. I don't like writing cards any more than you do. I definitely don't want you to feel pressured into doing anything you don't want to do. But it sounds like my mom is giving you a hard time about it, and thats not fair because she's my mom, and its clear what she's doing is stressing you out. I'm going to talk to her about it. If she bothers you again after I talk to her, can you just ask her to come talk to me if she has a problem?" (etc discussion)
"Thanks. I really appreciate feeling like we're a team"
"Me too"