Since this has been exacerbated by lockdown (like so many things) I’d focus on coming up with short term measures that could help. For example, could more school work be done with you, and less on the days with Parent A? If the two parents were still in a relationship, one of them would probably be more suited (in character and/or work circumstances) to do the home schooling, and an approach reflecting this across the two households could be helpful. Also, parent B has TV r emotional and practical support of a spouse, and parent B doesn’t - it’s really hard living through lockdown as a lone adult in a household and it might be helpful for parent B to kindly and gently discuss this with the child, in context of yes it’s easier for there to be fun at our house right now because I have a new spouse who can help with all the chores, and parent A doesn’t have that.
Eight is too young for the child to choose and although 50/50 often doesn’t work as kids get older, and needs adjusting, with all the extra uncertainty and stress of a pandemic it isn’t time to change that now.
Kids have a lot to worry about right now, that does show up in their behaviour at times. This child has extra to deal with, with his parents broken relationship, a new relationship for one parent, and living between two homes. So the fact that challenging behaviour shows up sometimes is understandable, and the fact that it happens with parent A and not the other doesn’t mean that parent A’s parenting isn’t as good or the relationship isn’t as strong; actually children who don’t feel safe or loved with an adult don’t habitually challenge them and their poor behaviour comes out elsewhere.
The stuff about the child’s feelings on the step parent situation is all a bit odd. Parent A is completely correct to say that the step parent isn’t a parent, and the child getting upset at that makes me wonder what they’re hearing at your house - he has Parent A (whether that’s mummy or daddy) and doesn’t need a replacement. Sometimes when children bring up things like this it’s because they’re actually trying to process the competing things they’re hearing, and struggling with competing loyalties. This might be doubly the case at the moment, lockdown means seeing one parent lonely and the other in a relationship - it’s a stark difference and sometime children then lean toward the parent who seems happier, avoiding their own complex emotions about the other parent. If this is the case then lots of understanding and compassionate conversation would be good, and potentially counselling over vid link - family therapy between both parents and the child would be great if parent A agreed to it.