It seems like such a basic question: what does the Father do? It is said that the Holy Spirit is the life-giver and speaks through the prophets and everything is made through the Son, Jesus, and Jesus is also the way by which we are saved and will judge the world. According to Trinitarians, what is the role of the Father?
The Father is the source of everything. (1 Co 8:6a).
1 Corinthians 8:6 English Standard Version (ESV) 6 yet for us there is one God, the Father, from whom are all things and for whom we exist, and one Lord, Jesus Christ, through whom are all things and through whom we exist
Contra your question, He is also the one who spoke through the prophets as per Hebrews 1:1-2.
He is also the maker of all things, and that through His Son at Hebrews 1:2.
Hebrews 1:1-2 God, having of old time spoken unto the fathers in the prophets by divers portions and in divers manners, hath at the end of these days spoken unto us in his Son, whom he appointed heir of all things, through whom also he made the worlds. (ASV)
You should read about the economic trinity. Basically, the Father plans, elects, foreknows, ordains, and sends.