I 110% disagree. This is someone's teenage son that has threatened to kill himself! Would you not want someone that knew that about your kid to tell you?
That would be the school or other authorities place...
Some other objective and qualified person.
Many parents (and I would suspect MOST parents) would just NOT take this kind of personal 'intervention' into their business very well.
I have BTDT....
When my son was younger, not even a teenager yet, there was a tentative friendship with this boy who turned out to be very troubled, possibly because of some developmental ASD type issues. When it became clear that this was NOT a good situation for my son, and he was calling my son all the time and trying to make my young child feel responsible for his problems... I actually did go to speak with his mom. I had met her a couple of times and felt that it might work out for me to approach her.
60 seconds after speaking with her and letting her know that I was worried/concerned about her son. I was ushered right out the door, and being called a 'witch' as I left.
ETA: As a small/petite woman, you just won't see me choosing to go personally, one-on-one, to go try to deal with this kind of thing with another parent. Whether it is a bully, a troubled kid who is controlling/bullying their girlfriend... whatever. I just can't generally advise anyone to do that. I never forget the old adage... "Don't like the message... Hate the messenger." There are usually other options.
OP, if you have any second thoughts at all about dealing with this parent... Just do NOT even consider it. Not all advice, no matter how well meaning, is good advice for any given situation.
If you do feel that this boy is troubled (and that would be my assumption) go to the school.....