Everyone in my family is vaccinated, except for a 2 year old grand daughter, and we’ve taken several flights in the last month to help one of our daughter’s move for a job change. My wife and I were vaccinated (1st shot of Moderna 2/28) and the other 4 in our family afterwards (J&J or Pfizer). Obviously we wore masks in the airport and plane per regulations, and when shopping or otherwise indoors but not when walking around (Los Angeles and Seattle).
None of us got sick and hopefully that will remain the case when we make the final move-in trip in 2 weeks.
If it was me I’d go visit your dad as long as everyone was vaccinated and other precautions that you mentioned were taken. Sooner the better I think given the declining effectiveness from the initial shots. Of course it’s a personally decision that only you can make.
As soon as booster shots are authorized or able to be taken I’m going in for one.