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My fathers very good friend is a partner at a BigLaw firm.

They are good friends because my fathers company has this law firm on retainer. My father is the COO of the company.

Is this a type of contact that can lead to a job or at least an interview? I don't even know if I am comfortable using this connection, but sometimes you gotta do what you gotta do.

This has been asked and answered many times, but in short: if your grades are around what the firm requires, this kind of connection will probably ensure that you receive an interview. If your grades aren't at that level, it depends on the partner's pull and how important your dad's company is. This connection is probably not enough to get you any more than the interview, though you should try to have a conversation with the partner so that in the interview you can say "I really love XYZ firm because of... I have also heard great things about the firm such as....from talking with Mr. Partner." Name-dropping, if legit, never hurts, so long as you can talk about something you discussed with the partner.