It's been a week since firm #1; is this wait detrimental at all to me, and if I call to accept firm #1 (fairly likely) can I just explain to the H.Partner (who gave me the call) that my parents wanted me to give it some time before I accepted? I'm feeling kind of pressured because I told them they were #1.
NO. Good grief. Please take this advice for your professional career, don't get into the habit of justifying things you don't have to justify in the workplace. The firm does not need to know what went into your decision process. And fer crissakes, leave your parents out of any conversation with the firm. You are not in kindergarten. You are a professional. Despite all the personal niceties wrapped around the hiring process, it is a business transaction
As for the pressure you feel, if H. Partner gets impatient, they will call you. No need to call them. When that happens, tell them that you unexpectedly received an offer that muddied the water and that you need a bit more time. (No "my parents think..." talk!)