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

I don’t believe that to be correct. When I was a Corporal E-4 Specialist attempted to give me trouble because they say we are both E-4 which is true but one is an NCO and the other is not. So in the first incident I teach. The second occurrence I bring the pain. I actually had way less issues as a SGT/E-5

Go ahead and parade rest for them. They are in a temporary leadership billet and their position should be respected. If they like chicken sh#t let them have it. Their temporary status can melt like a snowflake on a hot rock. They are on thinner ice than you.
MC
C co.
2/47th mech inf
RVN 1967-1968

Sounds like your squad leader has turned chicken-shit. Make sergeant as fast as you can. In the mean time go to parade rest etc. play the game until you can either PCS or outrank his ass.

Proper military courtesy is what it's called. That and they were promoted not only to SPC before you, but they also were selected as squad leaders. RHIP, the oldest term for you respect me and the position I fill and I'll respect yours. Even as low SPC on the totem pole.

Back in my day. You went behind the tank and handled your business. 2 men enter 1 man leaves to go buy the beer. But I’m just a old retired E-7. What do I know

I had this issue too, when I was a specialist. I was along the lines of "hey, you're not there yet buddy." But in reality, if they were promoted before you, and they are in a leadership position over you, they outrank you. You say you respect they are team leaders, yet don't give them the courtesy due their position. I am sorry to say it sounds like you do need, as I did, some corrective training.

I can summarize: I was an E6 when I finally caught a slot to go Airborne. I had and O-6 full bird in my platoon (student platoon sgt). When I called the platoon to Attn - he responded the same as the E-2 Pvt.

Doc Richardson,
Your question reminds me of a point in my career when I was assigned to an Infantry Stryker company. I was assigned to the MGS Plt; but because we were over staffed in E-5’s I was assigned to one of the INF. Plt’s as a Vehicle Commander. The way they did the CoC was that the Vehicle crews were assigned to one of the fire teams in each squad. The team leader of my assigned fire team was a Cpl and I was a SGT. It was discussed with the squad leader and the Plt SGT upon my assignment with myself and the team leader present just on where his “authority “ over me stopped. There was a line drawn in the sand so to say. Doesn’t mean that when he performed corrective training (ie. pushups) on the team I didn’t participate. I was a member of the team and by position he was in charge. There were points between him and me when I suggested that we take the issue to the squad leader to solve.
My point is when in doubt follow the order then get clarification from up higher as needed.

It's not the "Chain of Rank.". It's the Chain of Command.

Everyone posting a comment here is correct. Another thing to keep in mind is that everyone has a different leadership style. This specialist has been placed in a leadership position. He is obviously doing his best to assert this. Just because there is no regulation requiring doesn't mean he's wrong. Quit worrying about what's correct and do what is right. If the others on the team see you butt heads on this, they will follow suit. The correct thing is to take it to your ncoic and complain. The right thing to do, is to accept his position of authority and offer him the same respect you would give a full nco.