I think you don't need her permission to talk to your parents about whatever you like, and if you needed that space and time, well, you needed it. Most people would in your situation. If you are hoping for R I think that getting your parents 'on side' and sharing with them in a way that means they dislike or judge her is going to make that more difficult. There might be better people you can confide in. Here, for a start - but also perhaps a male friend, or a therapist?
You also don't have any control over whether she speaks to your parents or not. It isn't a case of 'letting' her do anything. She's free to do as she likes. Is she actually asking your permission? If so, I'd say 'you must do as you think is best' and then leave her to it - and don't ask her or your parents about their conversation. If she isn't asking your permission, I'd mind your own business.
As for helping around the house. The word 'helping' is a little odd. You live there. You eat meals and make laundry and dirty the floors just as anyone else does. Doing a fair share in the house - not as a way of nicing her back to you, or getting her to be impressed with you - but just because you are an adult and you live there - is a no-brainer. You looking after your share of domestic work doesn't need to have anything to do with your marriage. You're not doing her a favour, you're not sulking and refusing to do domestic work to punish her. You're just acting like all attractive, appealing men do - (right?) - independently doing your share of the house hold tasks.
What other steps do you have in place for your mental well being? What are your GAL plans? Do you have a male friend outside of the situation - not a friend of hers, not a member of your family or hers - that you can confide in?
