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

They are team leaders, which is an NCO position typically, give them the same respect you would a Corporal or Sergeant. Your example is what the junior soldiers will follow, if you show them a lack of respect so will the soldiers that look to you as an example. Dont be a blue falcon bro!

As an 0-3 I was fortunate enough to be selected for Company Command. My Company had many officers who outranked me including an 0-6 Dentist and many Senior Enlisted with 17, 20 and more years of service. I has no trouble Commanding the unit because everyone understood the difference between Rank and Position as I understood the incredible wealth of knowledge retained in those higher ranking Officers AND Senior NCOs. We all have our roll and function. Be part of the team. Know your roll and function and do it well.

Review your general orders, your enlistment oath and also address the power issue with your support chain. ‘I will obey the orders of those appointed over me’ sound familiar?

Don’t be that guy. Work with the team leader to make his or her job easier. Be a future leader and a role model for younger soldiers, who are watching and listening. Plus, when your in the leadership role, hurt feelings of those you scorned may come back to burn YOU. Be a Team builder.

Thankfully when I was active (1959-1962) enlisted just talked to each other and got things done. Always had the highest respect for my sgt's. They knew what they were doing and never let us get grief from the O crowd.

Assuming the position of parade rest every time you have a conversation with your team leader is a bit over the top. Using rank and last name are always appropriate. In any case, as the position states, the SPC is your leader and tradition and regulations dictate parade rest and addressing the team leader in the appropriate manner is required. In order to clearly draw the line on pay grade vice position, consideration should be given to converting the team leader to Corporal.

To answer your question: no, their is no AR (or FM) that requires a SPC (or PVT, PV2, or PFC) to stand at parade rest for a SPC, regardless of the position that SPC is assigned. Reference is FM 7-21.13, Ch 4.
Now, to the question you didn't ask: should you stand at parade rest for the SPC assigned as team leader? Yes, for most of the various reasons others have stated.
As a SFC attending ANCOC (SLC) I stood at parade rest for SSGs who were BNCOC (SLC) cadre at the NCOA. That was NCOA policy and I didn't have a problem with it there, or other schools I attended. I did it because it was the right thing to do, and because I wasn't going fo provide the opportunity for those few in such positions that seek out higher ranking NCOs ao they can "put them" at parade rest.
I have also been in units where PVT-PFC would stand at parade rest for SPC. Not a UCMJ actionable "offense" for failing to do so, but there were consequences (e.g., Iron Mike's).
There is nothing wrong with going to parade rest for your TM LDR, it demonstrates respect for the position, and builds your own self discipline.

You should be defaulting to that, it’s discipline. If you have a problem with it, be better than one of them and take their spot.

I heard of going to parade rest for NCOs. Long after my time. Stupid. What is this? Der Wehrmacht?

I agree with the others on here and respect their input. You have to remember that even though your team leaders are Specialists like you, they have been placed in a team leader position, which is a NCO position. Customs and courtesies go a long way in the military and you need to be the example for other soldiers to emulate. One day you could be put in that position and you will want the respect from your subordinates.

With them wanting you to address them as Specialist, I respect that of them. I will address other Sergeants who are my peers by their rank, even as a squad leader I still give my SGT team leaders that respect because they earned it. I had a 1SG who instilled that in me when I was a PVT.

Remember the 3rd sentence of the last stanza in the Creed of the Noncomissioned Officer: I will be loyal to those with whom I serve; seniors, peers and subordinates alike.