Writing in the present tense is always tricky. That's why most stories use past tense, but I'll help you rework it while keeping it present:
His father worries Henry has a death wish and asks the narrator, who seems to be his son’s old friend, for advice, but Henry reminds them of their father’s suicide."
--First of all, there is the issue of 'his father.' Whose father? I ask because this sentence alone almost sounds like Henry's, but that conflicts with the end with
'their father's suicide.' Clarify the pronouns to whom they belong to before implementing them.
-- I suggest stripping out the 'who seems to be his son's old friend' part out or shortening it down to 'his son's old friend,' and if you do so preferably also replacing 'his son's' with an actual name. ''Who seems to be his son's old friend' is clunky.
"Henry's father worries that his son has a death wish, so he turns to the narrator, an old friend of Henry's, for advice as the friend has also lost his father to suicide."