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

There is a good chance that higher up is telling them to correct soldiers for not standing at parade rest. Just do both of you a favor and just do it.

Respect the position. My platoon leader and sergeant had a meeting with everyone when I promoted into a team leader role before I got my stripes and told everyone treat me like an NCO regardless of what rank I had on my chest currently. I ran into a similar situation also when I was an E-5 Squad Leader with E-5 Team Leaders under me. They stood at parade rest for me because of the position, just like I stood at parade rest for the other E-6 Squad Leaders. Be the example for the younger guys in your squad and do what is right.

I'm reminded of the fact that billeting is covered under regulations... I'll use my own active time as an example (even though my command chose to ignore the regulation...) 1. When an enlisted member is assigned to a billet two grades or more above their rank, they are to be advanced to the next rank of junior NCO to designate their authority -- i.e. E-3 in an E-5 billet would become an E-4 Corporal (or in my case I filled an E-6 slot as the Battalion Medical Supply NCO, so I should have been advanced from E-4 Specialist to E-4 Corporal). This is mentioned in part in AR 600-8-19, 7-22.
In regards to this question of two individuals of same rank, it is ultimately the regulations regarding good order and discipline. Whether in formal formation or within visual observation of other soldiers, it is good discipline to show respect for position/billet. In most cases where there is not a formal directive or duty taking being undertaken (i.e. a work detail) there is rarely a case of conflict for two E-4s working together.

If he has been put into a leadership position then it doesn't matter that you are the same rank. If he gives you a lawful order, follow it. In the long run, you both are in positions to learn. He is learning how to lead, and you, to follow.
To get a better point of view, think about what you would expect from your squad/team members if the roles were reversed.

There isn't per se a an Army Regulation that states a specialist has to go to parade rest for another specialist. However, if one specialist is senior or has been authorized to act in a leadership capacity, like team leader or squad leader they can lawfully order the other specialist to a position of parade rest while exercising a command presence. Under article 92 of the Uniformed Code of Military Justice resisting or disobeying that order can result in punishment if supported by the chain of command. http://www.jordanucmjlaw.com/Articles/Article-92-Failure-to-Obey-Order-or-Regulation.aspx

Are you a team leader? Even though you are the same rank, they hold a higher position then you do.

In the short run just go with it. How are they as leaders overall? If being motarded with formalities is their worst sin you have much to be thankful for. I've known many leaders who came off as hard@$$ especially with formalities, but they were among the best overall leaders I have ever known, and I strive every day to be like them.