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

Don't confuse your rank with their position. I just left command of a 420 person company where I had numerous LTC's and COL's, but as the commander I outranked every single one of them by position. I didn't need to throw my position around, but ultimately my orders were lawful. Wherever there is a lack of manpower or overly competent personnel (every unit) you are going to have people in positions above their pay grade. Your situation is the exact same thing.
By virtue of position those specialist can give you lawful orders. Calling them a specialist and their last name rather by name only is an appropriate courtesy. Standing at parade rest if they tell you to, is also appropriate. Ultimately they are issuing lawful orders to you and you know the position they hold. If you disobey them, you would be punished and your defense wouldn't cut it. I do believe your company should get to work and make some of these SPC's into CPL's. However, it is a good test for them to develop as leaders and if you don't like it, get yourself in position to be promoted and you will have your own team. Not everything is in an Army Regulation, but trust me on this one. You don't want to find out you were wrong upon visiting trial defense services prior to your article 15.

SPC (Join to see) - Your post reminded me of this little riddle:
Q.) What you call a Captain-select?
A.) Lieutenant
Keep in mind that they hold a billet. So, both learn from them... and teach them how to be better SPC's. Sounds like they're trying to establish some ground rules for their soldiers. Sure, it might be annoying that you're the same rank. But, extend that professional courtesy. You'll be a better leader for it --especially when you assume their billet.

Specialist Richardson,
B.L.U.F
Stand at parade rest because its position over rank. However, if you want some insight continue reading. As an infantryman this is standard practice. This is an opportunity for both yourself and that team leader regardless of rank to develop both tact and resilency. There are going to be situations that you as the Soldier may or may not agree with, but you will execute them. They'll be giving these orders in their own name but remember they do come from higher. If outside of work you both carry a cordial relationship use the time outside of work to help mentor and mold if the opportunity should come up. If all else fail then simply put it in perspective and reverse the roles what would you do? Also taking into account who your peers and subordinates are then trying to ID the correct leadership style.

Perhaps this quote of General John J. Pershing would solve these questions if we just followed it.

“In a society where a man is placed in a superior position to another, if he is a gentleman, he never thinks of it. In the same society when a man is placed in a subordinate position to another, if he is a gentleman, he never forgets it.”

From the IG desk: Common military protocol requires being addressed by rank, then name. I am, to every military person I meet, to be addressed as "Colonel Hudson" regardless of the rank of the individual addressing me. However, this requirement may be softened in certain environments with the approval of all concerned. When faced with a situation of similar rank all around - then the individual with the oldest date of rank would be noted as the de facto ranking member IF placed in a leadership position. If conflict arises due to this understanding, then all participants should meet with their ranking NCO or Commander and hash the matter out. Your question as to "parade rest" lacks context for me to be definitive, but I am not aware of any requirement of two military individuals of equal rank to have one standing at parade rest while addressing the other, regardless of date of rank. During my career, I have seen such issues resolved by designating the E-4 in a leadership position as a Corporal with two 'hard stripes,' providing a proper visual recognition of that individual's position and responsibilities. This is also a matter to be resolved by your senior NCO and Commander if you feel you are more deserving of the position than the incumbent.

A Team leader in an Infantry squad is and normally reserved for an E5. The Specialist in this position is assuming that authority under the auspice of the Company commander. Therefore they should be respected and treated as a Sergeant as they are acting as such. Now they should be mentored and coached ensuring that they do things correctly. Lastly, the SSG, Platoon Sergeant or Platoon leader should have briefed your squad as to the state of the COC.

This was an issue back in my E-4 days. Bottom line, you respect the position and support your team leader. You may be in that position one day and you would want the same.

No actual reg stating you stand at Parade Rest for them. But they are your Team Leaders...therefore they are higher in the food chain than you are. Always remember the phrase "Position comes before rank." Might as well start standing that way now because apparently your CoC sees potential in them and they will more than likely make SGT soon....then you WILL have to stand at Parade Rest. You may not like the person, but you need to respect their position.

Schools often want you to stand at parade rest to instructors whom you may have time in grade or rank over. I pick battles that are worth fighting when it comes to things like this. You would like them to respect your position, then do the same for them.

I am not infantry by any means, but I thought that team leaders were Corporals. I think having the "hard" stripe would help in this situation. Having said that, I would just pick a better battle and play the game.