Should I ask my online PhD interview panel whether my partner can sit in?
I would find this odd and unprofessional. I recommend against this.
Also, I want to be able to talk freely and openly with you in an interview, and have you respond in kind, and I would be worried about how having your girlfriend on the line will affect this dynamic. (For example: I might worry that you would be distracted by thinking about how she is perceiving the interview, or trying to impress her.)
I might also be concerned that needing moral support for an interview this badly does not say much about your ability to handle the stresses of graduate school.
Consider whether or not this would be considered appropriate were you doing an in-person interview. Would anyone be allowed to sit in on said interview, even if they were told specifically not to interfere in any way?
Now also consider that even if your girlfriend doesn't verbally react to your interview, that her presence there, and her facial reactions to the interview going on (assuming you will have video) could change how you react during the interview.
Finally, consider whether or not this will make a good impression on your professor, or anyone else sitting in on this interview. Even if he is usually very understanding, he might find it a very odd request, and question the helpfulness of doing so.
If, and only if, you have a very good reason to have her sitting in on it should you even consider inviting her, and usually 'moral support' is not a good enough reason - this is meant to be a serious interview, and anything not contributing to it in a significant way could very easily be treated with some skepticism.
As some have suggested, recording the interview might not be frowned upon - even for personal reasons, but I would not share it with anyone else unless you have explicit permission to do so.
What is said during the interview could be very personal and private information that you should not be disclosing to others - which, by the way, is another good reason why this request would be awkward and unadvisable.
I agree with everyone else that having your girlfriend join in or even asking if that's allowed sounds very unprofessional.
However, as an alternative, I'm pretty sure there are screen recording programs out there. Perhaps you could ask permission to record the conversation? That way you can analyse it with your girlfriend afterwards.
If you still want to add your girlfriend to the conversation you should definitely ask the interviewers beforehand through e-mail and drop it if you have technical issues. But again, I would recommend against doing it at all.