I completely empathize with you. It is incredibly difficult not to cramp your eyeballs rolling them when your H suddenly turns into super dad. And, hard not to pin motivations on him, like trying to criticize or control you.
But. You need to let. this. go. Truly, what bothers you about it? That he didn't do it before and now suddenly is-- to which I would say, the past is the past, neither of you can change it, and if you take that history away isn't it a good thing that he's participating now and not staying home to play video games?
Or, you think he's doing it to criticize you... so what? Dissecting his motivations and getting angry about what you think might be why he's doing this is not a productive use of your time and energy. And honestly, while I can believe that a parent would be motivated to start this behavior out of spitefulness, I can't imagine them continuing it for any long period of time for that same reason. It is simply too much work. My guess is maybe it started that way but now he's enjoying the involvement, like the guys that have posted here on your board. You see some WAHs complaining about wanting more time with their children, especially over on the MLC board, but many times it is just for show.
My H has become far more engaged throughout this whole process in parenting and cooking (ha ha, unfortunately not so much the picking up or vacuuming) and it was hilarious to see him "discover" something I've known for years. But I also found that I was rather controlling in some aspects of parenting and the household, especially because he traveled so much for work. I let that all go. And truly, letting go of the emotional burden of being the only person to remember what days were swimming or soccer or piano and who has a spelling test on Thursday vs. Friday... O. M. G. I didn't realize how much weight I was carrying until he took over half of it.
You may need to also spend some time asking yourself (as I did, at first) why this all bothers you so much. For me, I felt a little displaced from my comfortable role as the preferred parent and unwritten but acknowledged keeper of all knowledge for the household. What did that mean for my identity? Was I no longer needed in the same way? Etc. But I got over that. (Now if AP tried to swan in and spend time with my children, that is a whole 'nother story and I still would want to rip her eyeballs out. Trying to deal with the rage I have right now surrounding her.) But my H stepping up as a father is a good thing.
However... I digress and his motivations for starting or continuing the behavior are truly irrelevant. Having an involved co-parent is not a bad thing. Who cares if he's touching base about the school supply list. Let it roll off your back and try to stop interpreting it as competitive or controlling. Take it for what it is and let the rest go.
((RM)))
Me (46) H (42) M:14 T:18, D9 & D11 4/19 - 12/19: series of escalating BDs 9/20 - present: R and piecing