If a non-NCO is a team leader, do you have to stand at Parade Rest when speaking to them?

No, there is no regulation that requires it. That said, "common practice" does vary from unit to unit, and sometimes you have to make a decision for yourself about who is and is not able to make your life miserable. My suggestion would be for you as a brand new private to go with the flow and do as your team leader asks, as well as address him by his proper rank. The rank decorum is especially common in an Infantry unit. When you have an opportunity to ask your squad leader or PSG whether you're really supposed to do that, then by all means, do so...and do that in a casual manner, as if you're really trying to learn. Again, my best advice for a brand new Private is to go with the flow and learn everything you can. Soon enough, you may be in a position where you are trying to establish yourself as a leader, and in the case of a SPC, try to keep in mind that he probably just got put into a position where he's now in charge of his friends, so he has no choice but to figure out how to draw some lines.

Are they a SPC or a CPL? Remember the Army sees CPL as an NCO. I am in a TL spot and would never expect nor allow a lower enlisted to stand at parade rest when speaking to me unless it was matter of formation. When I pin CPL or SGT, I will expect them to, but for now, I do not expect it, nor is it a custom reg.

There are a couple of ways to answer this. On paper, probably not. In practice in an infantry unit, I have seen it enforced zealously. I don't necessarily agree with the practice, but best not to sink the boat while you are standing in it.