Do they know the google account password you use for the phone? If not, then they won't have access to your email or be able to remotely install apps on your phone (I assume that you are not worried about physical access to the phone, because in that case it wouldn't matter whose name the phone was in). Your father's name in the billing system won't mean that he has access to data on the device unless you have given him that access yourself (the phone contract and the Google account you put on the phone are entirely separate).
Depending on how your provider works, what sort of contract you have and what billing options are available, he might be able to see who you have been calling or texting, but not the content of the messages. I have fully itemised billing, so I can see online who I have called and who I have sent messages to (here in the UK incoming calls and messages are not charged, so do not appear on bills - I don't know where you are, so things may be different for you). I can also see how much data I have used, but not what it has been used for. But again, that depends on your network and contract type. My daughter has a pre-paid phone which is in my name, but to get that type of billing information available to me requires a set-up process in which the network will send a code to my daughter's phone, so at least on my network I could not do that without her co-operation (or without my having access to her phone). I may even have had to do that to set up my own online billing account - it was a long time ago - but when I received paper bills they were itemised anyway. Who are the bills sent to in this case?
In short, unless you've given him access to the phone which could be used to install spyware, the fact that the phone it in his name does not give him access to the content of your communications. But it's possible, depending on your network's billing options and how the account is set up, that he might be able to see who you have been calling or messaging (at least their numbers, not their names). I cannot give a definite yes or no to that last point.
