@ccero-san Thank you for bringing up this issue. All other preceding answers including mine focus on the case when we use 父上(chichiue) when addressing our own fathers.( which is why we say we rarely or never use it these days.) That ‘s because ( I guess ) when we heard the word “父親”, our own fathers instantly come up to our minds.
As ccero-san says, there are surely some particular occasions where you are expected to use 父上 (chichiue).
I think that how often you use it in daily life could vary by gender, age, social standing, where you live, how you want to sound, your preference, and all other possible factors that you can imagine.