Have your mom come vist and talk with her about their situation. It's both new to them and no one quite prepares you for the emotional side of retirement. Retirement is a bit like the first year of marriage. It's a HUGE adjustment for both.
My mom "retired" when my dad did. So Dad now loaded the dishwasher and put away for her. Mom still cooked but dad did his share of cooking by taking her out to dinner!! Dad also got involved in the Masons again...he had stopped going because of his work schedule and being busy with us kids.
He also volunteered and worked out three times a day by either walking or swimming. Whenever he got on Mom's nerves, she found something around the house that needed fixing or adjusting or painted, cleaned (cleaning air filters works well!) or she just went shopping at the local mall for awhile LOL! Whenever Dad got frustrated with Mom, he went and hung out at the hardware store or went and got a haircut or went to the library to research the family tree.
It's true that when men retire they feel somewhat useless. They are used to having a purpose and interfacing with people everyday. So, it's best to try to recreate that environment - senior style. If you or other siblings live close by, involved Dad in a project - , painting, fixing a leaky sink, changing the oil in the car, remodeling a bathroom, shopping for a ceiling fan, whatever dad's good at.
When mom comes to visit you, have her create a once a week "honey do" jar. Fill it with slips of paper with odd jobs that need doing around the house. Once a week Dad picks a job from the 'honey do' jar and does it! That's his project for the week. You can make a jar for day trips, or for restaurants.
Travel, go see a movie, tour a museum, go to a ball game, volunteer somewhere, deliver meals on wheels, take a class at the local community college or university, try something you never thought you would.
It's hard when you're parents aren't getting along but please realise that this is their problem that they need to sort out for themselves. You're not guilty of anything but loving your parents.