Jesus Christ did not assume the position of the Father. As God who did not consider equality with God something to be grasped (Phil 2:6-8) and who shared our humanity (Heb 2:14), Jesus identified himself as the Son and called God His Father. But we should not go beyond the Bible and say that He is not the Father.
We can’t try to think of God in human terms, because God is spirit. We have to acknowledge that the relationship between the Father and the Son is beyond human reason.
How could He be in three places at one time during baptism?
God is omnipresent: He is present everywhere. He is beside you right now, at the same time as He is present on the other side of the world. Even though as humans we cannot conceive how God can be at many different places at the same time, we can’t limit God to our human concepts. He is Spirit, and He transcends all physical limitations.