You both sound a little immature as a couple. Is the marriage young?
Since you say you want to remain married, the only way forward is for one of you to bite the bullet and compromise. Since you posted and seem more emotionally aware of the situation it should be you.
The answer is to drop the hatchet and be especially nice to his parents. Instead of going exclusively to your parents every Saturday, could you take the initiative and visit his on Sat. without prompting? It would be hard and you may feel disloyal, but you aren't married to your parents. You're married to your husband and your marriage and relationship with him is your first priority.
You have both chosen to take on your parents feud and have not set up boundaries. By that I mean a commitment that you will both not allow yourselves to take sides to the detriment of your relationship. You can both love your parents without taking sides to an extent that encroaches on you both and undermines your marriage.
So work out a way where you can see his parents more, and be genuinely good to them, and maybe he'll soften towards yours. Also reconsider how much time you spend on parents vs yourselves as a unit.
Also, never say anything negative about his parents no matter how negative he is about yours. It may involve a lot of patience, and biting your tongue.
You and DH should see yourselves as a contained unit separate, but involved with both sides.
Rather than taking sides, be a bridge between them.