Jesus is, according to his own words, a man. He is not expressed as God anywhere so when he died, he remained dead until the Father, his God, raised him.
now you are trying to kill me, a man who has told you the truth that I heard from God. John 8:40
Jesus is still a man - though a man with spirit, eternal life at God's side.
For there is one God and one mediator between God and mankind, the man Christ Jesus. 1Tim 2:5
He has set a day when He will judge the world with justice by the man He has appointed. He has given proof of this to everyone by raising him from the dead.” Acts 17:31
(Many think scripture says Jesus raised himself. It does not, but by reading theology into the text, it can be made to say anything. When striving to understand scripture with our own wisdom, we mustn’t generate conflicts with other scripture. Jesus raising himself, even though he was dead, generates many conflicts and is thereby rendered an invalid construct.)
The Father didn't die - He cannot die, He is immortal. He is the only being who is immortal. 1Tim 6:16 Some say the son is also God, but this is not of scripture.
Jesus was not immortal - that's why he could die. But once raised from the dead, God made him immortal too - unable to die anymore. Jesus was born of Mary about 2000 years ago as the Gospels express. He did not exist before then except according to God's prophecies starting from Gen 3:15
He was put to death in the body but made alive in the spirit. 1Pet 3:18
We know that since Christ was raised from the dead, he cannot die again; death no longer has mastery over him. Rom 6:9
The Holy Spirit is not an entity that can die - it is the spiritual expression and power of God the Father just as His 'word' (logos) is. John 1 and 1John 1 (the logos isn't a person or entity either that it could die)
All things have been handed over to me by my Father, and no one knows who the Son is except the Father, and who the Father is except the Son, and anyone to whom the Son wills to reveal Him. Luke 10:22
We see that if the Holy Spirit is part of a trinity, he would know both the Father and the son. This verse expresses the deepest intimate knowing that the Father and Jesus share - but no mention of the holy spirit. Many want to ignore this verse or read-in that which they feel is missing - we do not need to alter the word of God to fit our favourite theology. We are warned against doing so, but this hasn't stopped many from adding to God's word!
There certainly has been a lot of confusion over these matters. The bible is quite clear if we choose to accept what it says without reading-in additional to what God has provided.
Only Jesus can die. He was born for this purpose and is the son spoken of in John 3:16
For God so loved the world that He gave His one and only son, that everyone who believes in him shall not perish but have eternal life.
Regarding Matt 28:19. This is not talking about God dying, but through baptism, we may be joined to Jesus who died for all people. He died that we would not have to - speaking of a spiritual death not a physical death that all die. "we have been united with him in a death" as your passage expresses.
Conclusion: When Jesus died, only he died. The Father cannot die and the Holy Spirit cannot die either. We are baptised “into his death”, the bible states this unambiguously.
Edit: regarding baptism in the Q, no NT baptism was ever done according to Matt 28:19. It was always in Jesus’ name only. Yet, there is no downside to recognising the Father who also enabled salvation and the spirit by which we may live again as Jesus did. We can read in more to this than we should as this apparent direction was not followed by anyone being baptised by Paul or anyone else. The name represents the power of salvation which is from God and His son Jesus through the spirit of life.
The works that I do in the name of my Father, these bear witness concerning me. John 10:25
Interestingly, the Holy Spirit has no name.