If you set things up the right way, mom only resting, holding baby and feeding for the first few days is feasible, and may even be ideal for establishing a good milk supply if she wants to exclusively breastfeed.
The first week or so, a breastfed newborn's basic needs are as follows:
- Clean diaper and clothing
- Nursing on demand (which is usually very, very often 'round the clock)
- Remaining at the right temperature
- Being held close (ideally skin-to-skin) by a parent/caregiver as much as possible
That's basically it. Newborns don't need to be bathed daily, and some of the household chores can wait a couple days for your wife to begin her physical recovery from birth. If you prepare in advance with a safe sleep space for baby in your bedroom, food for the adults ready to easily heat up, plenty of diapers and baby supplies stocked up, etc., it's reasonable to let mom rest, nurse, hold the baby and have other family members take care of the other tasks, which are essentially, diaper changes, making sure the new mother is fed and drinks enough water, care for the dog, and washing lots and lots of baby laundry. You may also be the one to get the baby in and out of the crib or bassinet for the first days if it's painful for your wife. Reaching down into the crib to pick up the baby was excruciating in the first days after my C-section!
Do be aware that a newborn is going to want to be held and/or nursed pretty much 24/7, and it's healthy to provide this level of contact as much as possible. So in this scenario your wife is probably still going to be the one caring for the baby a good deal of the time, but 'caring' in this case is mainly just holding/skin-to-skin, snuggling, rocking, nursing, nursing, nursing some more.
It's probably not a feasible plan for very long, as the needs of a baby rapidly expand, but as you have no other children to care for you may be able to make it happen for a brief time while she recovers.
Having your parents to help or not is a very personal thing. It's not uncommon for a brand new mom to be very 'territorial' and not want any extra people in the house, and especially not to want to be a hostess for any guests. Also some grandparents are pleased to wash out poopy baby clothes and make a run to the store because you forgot to buy baby shampoo, and others would like to be entertained, hold your new baby, eat whatever is in your fridge and generally be more work than help. Only you know which kind you have!
If your wife does not want to have your parents there right away because she honestly doesn't think it would be actually helpful, I think it's best to go along with her wishes.
Edit: For full disclosure, I did not stay in bed/my room after my baby was born, but he nursed so long and so often that I was basically in the rocking chair or on the couch and out of commission at least 50% of the time anyhow for about the first 5 days!