This sounds very similar to my relationship with my Mother. I sent cards etc for Christmas, she didn't send anything for DC. I'd be less bothered if she'd simply ignored me, but doing that to my DC felt quite unpleasant.
Like your Mother, she wasn't particularly affectionate as I was growing up. She could also be quite emotionally manipulative and didn't appear to have a great handle on her stress responses. If I said I missed my dad or wanted to see him at the weekend following their divorce she'd say I was ungrateful for everything she'd done for me and generally make me feel guilty about wanting to spend time with him.
Like you, if I don't do as my Mother asks then I am in for months of being ignored and ostracised. Just a small example, I went on holiday recently and a friend asked me to get her some cigarettes. I agreed. My Mother then asked if I could get her some and I said my customs allowance was already taken up by the friend's request. The result was lots of passive-aggressive Facebook memes, pictures of my siblings saying "this is all the family I need", and the current stint of ongoing silence.
In all honesty, there is no easy answer to this. It sounds as though not having much contact for your own sanity could have an effect on your relationship with your Dad as well? For me, this was never an issue (because of the divorce).
I'd highly recommend finding a counsellor who has experience in helping people who may want to go No Contact (NC) with a parent. They don't encourage you to reach that point, but they do help you explore the absolute sh*t storm of emotions that comes with being in your situation.
My stance has been to maintain minimal contact. I don't do any chasing and I don't pander to her desire to cause problems and then act like a victim. At the same time, I don't tell her to go away if she does get in touch. I'm not sure what would be right for you, but this is what causes the least drama for me and my DC.
It's genuinely a naff situation to be in. Having your own kids can make all of this bubble to the surface. Really give the counselling option some thought, because unpicking your emotions surrounding this by yourself can feel very frustrating and lonely!