May my father purchase Mishloach manos for me to give to my friend? Would I fulfill the mitzva that way?
Yes, this is fine.
By contrast with a number of mitzvos, such as the Four Species, we don’t find a formal requirement for mishloach manos to belong to the giver.
The principle of the mitzvah is that one person should send foods to his fellow, and this is fulfilled without the formal principles of ownership.
See, for instance, Darchei Torah 695, where we find that a person fulfills his mitzvah but inviting others to his home – even though he doesn’t transfer ownership (though some Rishonim (in Nedarim) maintain that ownership transfers automatically to guests who eat at one’s table). Likewise there is no need for a formal transfer of ownership to a son.