Is there anything I can/should do about an NCO punishing a SPC Team Chief in front of his soldiers?

"By stupid stuff I'm referring to not standing at parade rest when speaking to a NCO"
At that point I stopped reading.....
A: Thats not "Stupid stuff"
B: That the poster feels it is, makes suspect any other observation being related here.
C: As the phrase says, "Stay in your lane" this is not your fight.. If you think the action your observing is immoral or illegal then you need to speak up to someone senior to all three of you, but other than that, not your business.
D: If you feel now, and still feel later when you have some experience and are put in a leadership position that the way that NCO handled a specific situation can be done better, use that memory and do it differently

and an Edit add on.. " NCO punishing a SPC Team Chief"

Enlisted generally do not have the authority to punish another enlisted SM. They are charged with and have the authority to take corrective action.
Punishment is reduction of rank, loss of pay or liberty, or the like.

Corrective action can be a verbal rebuke, detailing what was wrong, what IS right and what will happen if right is not done next time. It can be reenforcement training related to what was done wrong.
In years past, (because it was effective at the time with the quality and type of SM we had) physical negative reenforcement was common, because it worked with the SM we had in service then. None of that is "punishment"

This is not your issue to fix. The SGT and SPC need to work it out themselves. Since you are a PFC there is nothing you can do about it anyway. Just remember what you are seeing. Things like this can serve as great examples of what not to do when you finally pin on NCO stripes. All leaders good or bad are beneficial in developing your leadership style.

Generally praise in public admonish in private is the way to go. However there isn't a lot to go on here. "Getting on him" about the stupid stuff can be interpreted in different ways. Something like parade rest displays discipline and respect. What you consider "getting after him" could very well be simple on the spot corrections. He may be a young SGT learning and developing his own leadership. Not saying your wrong but the working dynamic in your section may not work the same with other individuals involved. I am hesitant to really go into depth with the limited information here. I would however advise you to be very careful when intervening on behalf of someone else. You may not always know all the facts even though he is your friend.