Hey,
My dad passed away too super suddenly, we had pretty much no warning, he was here and then he wasn't anymore. It is now two and a half months since he passed away. I had and still have so much anxiety, panic attacks, nausea, brain fog and am still processing it.
I am studying and couldnt imagine trying to work through everything. Even with studying I can only manage half days and about 2 or 3 days a week. If your work is supportive, they might be open to reducing your hours/days/tasks. I think if there is a safe and quiet space you can go to while youre at work that might be helpful too if you need some time out.
I dont think parents should pass. Its so unbelievably hard and traumatising.