Sorry for your loss. My dad also had PD and died last year. Like you, I wasn’t able to be present when my dad died.
Everybody’s different, bit I think I was helped by participating in many of the conventional grief-related rituals. My siblings and I wrote up memorable obituaries for each of my parents, and I wrote a more personal (to me) version to share with some close friends. When my mom died several years ago, we had a memorial service and luncheon, and as a family we scattered some of her ashes in a wilderness location. When my dad died, we had a private military commitment ceremony and then a memorial luncheon with friends, and scattered some of his ashes in that same wilderness location. Before any of these things came into my life they all seemed like just unpleasant social obligations, but looking back, I really do think they were important for my own healing. In retrospect, I think “unpleasant” isn’t really a good word to describe these things. Crying your eyes out while you’re trying to write an obituary may be emotionally painful, but it’s also deeply meaningful.
And speaking of crying, give yourself permission to do that. If you don’t feel comfortable being in such an uncomposed state around other people, then find some private space where you can let all that sadness pour out without any inhibition. Consider taking some time off of work if you can, and let your coworkers know why. I went back to work too soon after my mom died, and had never told my coworkers why I was gone at all in the first place; when I came back, I ended up falling to pieces when a coworker asked me how my weekend went.
I gather you were close to your dad. Since he was the first of your parents to die, this is a new experience, and your uncertainty about how to move forward is understandable. In the months before my mom died, I was kind of “pre-grieving” because I knew she would die soon, but I was also scared of what the aftermath would be like. It was bad of course, terrible grief, but people do heal and get on with their lives, and so did I; it just takes time.
My siblings and I were helped by a family friend who wrote to us after my mom died. He had lost his wife to cancer a decade prior, and said that for some time after she died, thoughts of her made him sad. But over time, he eventually got to a place where thoughts of her made him happy instead. It may be helpful for you to keep that in mind. Other people have lost their parents before you - including your own parents! - and you can see that even though they may have experienced tremendous grief at the time, they were able to eventually get back to a sense of peace and calm.
Having said all that, I find that a sense of loss still comes to the surface now and then. Example, Christmas crackers are an English tradition from way back, and my mom (born in England) kept it up with all of us until she died. At Christmas a year ago, my wife and I were unwrapping a box of crackers, and thoughts of my mom suddenly sent me down a rabbit hole of tearful grief, even though it had been many years since she died.
Some people have a harder time getting back to normal. Talking with friends can help (and strangers too - hopefully this thread is useful for you), but if you find yourself really struggling, you might consider seeing a professional therapist. With their training, they can be better than friends and family at helping you untangle complicated/painful feelings and work through it all in a productive way.