How safe is nursing home respite care?

Thank you for taking care of your grandmother. Caregiving for those we love is challenging, but rewarding. I understand your concern about COVID. Facilities are being more pro-active about testing their staff and managing the virus. Both respite care and in home care are usually paid for privately. You can ask the facility if they would take Medicaid for a short term stay. I would recommend you look at a facility for respite because if she is still at home, you are going to feel like you still have to manage her physical care even if you bring in support. Does she have long term care insurance? That could also be an option to help pay for a respite.
It's not wrong to make self-care a priority. Imagine what will happen if you don't take care of yourself and something happens to you. Who will help her? If you can answer that question with the name of a person, ask them for help! Sometimes people want to help but don't know how. All they can say is "no".

That question depends on how much omicron is in your area, how seriously the general public are at mitigating spread (staff and their contacts are the weakest link) and how stringently safety precautions are enforced in each facility. I would also hesitate to place her in a facility if that would mean she is required to quarantine for an extended period of time after her arrival and I would want to know whether the facility is overwhelmed with staffing shortages because either of those things could make her stay very unpleasant and possibly contribute to a significant decline in her physical and cognitive functions.
As for fear of getting the virus - I think that having her fully vaccinated is likely the best you can do whether she goes to a facility or you choose a home care option, ultimately you have to make the decision based on your personal risk/benefit analysis.

My 86-yr old MIL is in LTC and got the original covid back in May 2020 and survived. Now, the NH has been boosting and testing and wearing PPE ad nauseum, Regardless of these precautions Omicron is still being transmitted to residents from staff and visitors, however the degree of severity is more like a cold and no one has required hospitalization and there's been no covid-related fatalities.

Your Grandma has Parkinsons, and I'm not sure where that puts her in term of "compromised" health, or if she's obese or has other medical issues, but eventually everyone will get Omicron, wearing a mask or not, boosted or not, unless you literally never leave your home or come in contact with other humans. If you go out into the world, or have family or aids coming in, eventually it will be brought to her.