According to the Tom Paris page at Memory Alpha
He was assigned the additional job of medic for The Doctor when an assessment for medical replacements was brought forth, as Paris had taken two semesters of biochemistry at the Academy.
Further, in the episode in question, the following exchange takes place
EMH: I see. I realise your Sickbay training is purely voluntary, but you haven't spent any time there in several weeks. And when I try to find you, you're always in the holodeck.
PARIS: Correct me if I'm wrong, Doc, but we haven't been treating anything more serious than an ingrown toenail.
Later during the conversation you mentioned with between Paris and Chakotay, Chakotay says
CHAKOTAY: Maybe. I was wondering if there's something on your mind. You're showing up for your duty shifts, you do what's required, but your heart doesn't seem to be in it. You seem preoccupied.
The first part indicates that he has been assigned to sickbay in some capacity but any training he gets from the Doctor beyond his initial academy training is voluntary. The second part indicates that Chakotay is concerned about his officer as he sees what he believes is a troubling pattern of behavior in Paris.
Also, he may have volunteered for something but once he did, it became his responsibility.