I'm looking to help make a promotional video to promote a friend's new company
Is there a legal eagle on here who could tell me if I can write new lyrics to a copyrighted song, or do I need to seek permission from the artist who wrote the original song?
I'm no legal expert but I would assume both the music and lyrics are copyrighted and since your intended use is commercial, you'd be wise to seek permission (in writing).
You are legally entitled to use any song, be that melody or lyrics as part of a parody or critique, but it is absolutely against the law to then use it for any form of advertisement, unless you get an agreement from the artist, song writer, rights owner and studio.