Is there an AR stating that a Specialist has to go to parade rest for a Specialist?

There's no AR, but your best COA is to go to parade rest due to position. It's going to cost some pride, but it is the right thing to do. Your other COA involves IG or command. Probably not what's best for you, your platoon, or the company as a whole. Pick your battles wisely

Team Leaders? Are you sure you're not working for Target? I was assigned to 10th MTN DIV, 4-31 INF. The unit started promoting more specialists to CPL/E-4. We had two and those we had to treat as NCOs (junior). When I got there I was an E-2 and by the 2nd deployment, there were E4/SPC who had longer TIS and TIG. We did not have to stand at anything for them. I think what is going on over there is ridiculous and doesn't follow rank.

I was once a 1LT platoon leader, with a 1LT company commander. I saluted my commander, came to attention, and addressed him a sir.

It’s not just about rank.

Used to be that we would laterally appoint E-4s in this situation to Corporal and be done with it. I guess this has fallen out of fashion.
Regardless, the Team Leader is in charge of you, and if that is how he wants to run his team then that is how it is. He needs all of his troops to respond to his orders with discipline and adroitly, else shit goes sideways when things need to get done under duress.
Choose the hill you want to die on, but this is not it.

The infantry has changed, I guess. Y'all act this way in Afghanistan? Is this the way the rest of the army works? Well, I have noticed little references to the Vietnam vets not being from a professional army, but this smacks of a bunch of martinets with only tangential combat.

Wow -- this happen to me back in the day -- Fort Lewis, WA -- I Corp HHC, and there was another SPC giving me instructions and telling me to go to parade rest, I was also SPC and at the time thought that was weird and inappropriate. Little did I know rank is one thing but TIME IN GRADE is another, any SPC with more months/years than me technically outrank me AND of course by position, as some of the other comments here have suggested. No worries, keep your head up and believe me your time will come. Good luck!

My advice to you.Lead by example .play along.one day it'll be you're turn. Kill with kindness.

I have seen this before when my Plt (2nd Plt B. Co. 1-187th In. Leader Rakkasans) was training up for deployment to Afghanistan (OEF 10-11). We had a Squad Leader (E-5) and two Spc. Team Leaders and we had to stand at PR for each of them. I was a Spc as well at the time with more time in service/ grade than both put together (prior service). It wasn't to offend/ disrespect anyone, but to teach the lower enlisted who weren't TLs respect for both Rank/ Position.
I also had to deal with this same arrangement when I was being Med-Boarded from Fort Bliss, TX. We had a PFC who passed SFAS and was promoted by the PSG to Team Leader and we were told that even if we were a Spc who was not a TL, we would have to stand at PR for this kid. Needless to say I was pissed because I didn't agree with the arrangement and because of my experience in combat, which of course, this kid (19 years old) had been in for maybe 1 year.

You do have to stand at parade read because that specialist is in a leadership role. I have asked that same question because when I was active duty, I was placed in a team leader position. I also was required to inspect my soldiers along with their living quarters and counsel them monthly or when they screwed up.

This is probably something that is expected from within your command. Any place that I have ever been (Army Aviation) this was not a thing. Example: our battalion CSM would never have stood at parade rest to address the brigade CSM.