VogelVogel this situation sounds really hard for all concerned. I am very sorry for you because I think you are all kind of stuck in a bad situation.
I would say if you want to see your parents, then you should do this.
I can understand why your wife would not like the children to see your parents after what they have said about her and the things in general.
I think in your shoes I would just see your parents, explain that you will not be bringing the children (when they are older they may decide to come and may be able to come with you). If your parents complain, remember they have created this situation.
And I would focus on your marriage and what you want to do. See if your wife will agree to some counselling together and work through some of these issues together.
You have not been as supportive as you could, you've made mistakes. You can't see the problems. But I do think you are both very unhappy.
You only get one life and you sound very miserable so I would focus on seeing if you can make your situation better, support your wife and your parents and try and do this all separately. Do not try to work it all out so everyone gets on together, because they simply will not.
I think you've been (I hope) honest about your own failings so you are at least aware. Are you aware how upset all this must have made your wife?
I think you and your wife need to work this out together, you have chosen to stay together so maybe there is still some love there, or maybe neither of you wishes to disturb the kids, or maybe you both do not want to rock the boat.
Your mums offer to look after a three-week-old baby was not going to go down well. Maybe this is a cultural thing.
Your mum saying she would take the kids away, was never going to go down well.
Your mum 'joking' about the paternity of your kids, was never going to go down well.
Sadly, this has all compacted to create a really horrible situation.
You don't need to sort all this out in one go, these are long historic issues. You just need to focus on doing what you feel you need to do for your wife and desperately, for your parents. I lost my dad 12 years ago and my mum last year and I know that when parents are ill it is very hard.
An affair will not help you. Work on the marriage or work on finding happiness apart, if you think this is what you want.
All the best.