Can a Specialist make a Private stand at Parade Rest for him?

I would check out TC 3-21.5 specifically the Rest Positions at the Halt. Parade Rest Note:
“Enlisted Soldiers assume this position when addressing all noncommissioned officers or when noncommissioned officers address noncommissioned officers of superior rank.” Seems pretty clear to me. SPC is NOT an NCO and is therefore not due this courtesy.

When I was a private I was required to stand at parade rest for anyone who outranked me. This was also when we would take our differences to the woodline rather than the counseling packet.

You have no military authority below CPl. SGt. unless given so by th unit commander. You have to be acting in the capacity as a supervisor. Often units give that authority by virtue of the commander I general terms. In either case do not assume, verify with the 1SG. Thank you for your service.

I held every rank from E-1 to O-4. When anybody called attention, I put them at ease as soon as possible. I appreciated the respect, but didn't "get off" by making them stand to attention. I think the soldiers or airmen appreciated this. We were there to do the mission, not impress any one.

SPC Elijah J. Henry, MBA When I was private and I was approached by a Specialist military bearing kicked in and I automatically went to Parade Rest and I continued that practice during my entire (9) years of enlisted service when reporting or answering to all those who were above me in rank (as a common courtesy) and even as a commissioned Officer up to including the rank of Full Bird Colonel. Maybe I was brain washed at the early age of 17, but I respected the Army, the Institution, and my immdediate leaders, and the country we served. It was about a higher calling for me. Don't care if there is a regulation or not. It was about soldiering and setting the example for others to emulate. That's my two cents, now I can stand at ease.

CHAPTER 5: MILITARY CUSTOMS & COURTESIES

PARADE REST
Like “Attention”, Parade Rest is a form of respect given to NCOs by those junior in rank. When a Senior NCO enters an area of junior enlisted soldiers or cadets, the room is called to “at ease”. All personnel should immediately go to the position of “stand at-ease” until told to “carry-on”. Likewise, when speaking to an NCO, soldiers junior in rank or cadets, will maintain the position of “parade rest” until instructed to “stand at ease”, “at ease”, or “rest”/relax.

Customs, Courtesies, and Traditions

The Army is an organization that instills pride in its members because of its history, mission, capabilities, and the respect it has earned in the service of the Nation. A reflection of that pride is visible in the customs, courtesies, and traditions the Army holds. Adherence to them connects us with soldiers throughout America's history.

OTHER COURTESIES

4-16. Military courtesy shows respect and reflects self-discipline. Consistent and proper military courtesy is an indicator of unit discipline, as well. Soldiers demonstrate courtesy in the way we address officers or NCOs of superior rank. Some other simple but visible signs of respect and self-discipline are as follows:

When talking to an officer of superior rank, stand at attention until ordered otherwise.
When you are dismissed, or when the officer departs, come to attention and salute.
When speaking to or being addressed a noncommissioned officer of superior rank, stand at parade rest until ordered otherwise.
When an officer of superior rank enters a room, the first soldier to recognize the officer calls personnel in the room to attention but does not salute. A salute indoors is rendered only when reporting.
When an NCO of superior rank enters the room, the first soldier to recognize the NCO calls the room to "At ease."
Walk on the left of an officer or NCO of superior rank.
When entering or exiting a vehicle, the junior ranking soldier is the first to enter, and the senior in rank is the first to exit.
When outdoors and approached by an NCO, you greet the NCO by saying, "Good morning, Sergeant," for example.
The first person who sees an officer enter a dining facility gives the order "At ease," unless a more senior officer is already present. Many units extend this courtesy to senior NCOs, also.
When you hear the command "At ease" in a dining facility, remain seated, silent and continue eating unless directed otherwise.

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I would love to see the regulation or policy that requires a Private to stand at parade rest to a Specialist. I'm not aware of one. Maybe there's some unit policy somewhere, but as far as I know there isn't anything in any Army Regulation that requires that. I don't think I ever seen that happen since basic training when the recruits in my platoon was treating the Unit Supply Specialist to that honor.

I'm also going to agree with SSG Therrien. By regulation, itis for NCOs. In the units I've been in, Signal Battalion we did not have privates stand at parade rest for SPCs, the MI unit we did not have privates stand at parade rest, and in my infantry unit, they went to parade rest for CPLs, not SPCs. I have seen, and I think it was out of habit, many of the 11Xs still went to parade rest for SPCs. So yeah, I've seen it done by the unit. It is also done if the SPC is in charge (head of a formation, etc) or appointed by someone with the proper authority so they are representing that person in which case parade rest would not be out of line. Require though, would be based on the unit and if their command has said such.

Interesting question, my friend SPC Elijah J. Henry, MBA
In my humble opinion only NCOs [and commissioned officer] can direct a soldier of lower rank "to stand at parade rest for them."
The only exception IMHO would be in a training situation in BCT or AIT where a SPC was striped based on duty position and not rank.
In my enlisted experience [1974-1976], only NCOs could direct me to parade rest, attention, or any other position, I was a combat engineer and later a cadet candidate as a PFC. I received a lot of instruction on customs and courtesies. :-)
Thanks for mentioning me COL Mikel J. Burroughs
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All I can say is go to the Ranger Battalion as a private and tell a “tabbed E-4” you ain’t standing at parade rest....then watch what happens!

In my first unit, we had a soldier who thought he could just boss anyone who didn't out rank him around. He was about 4 months ahead of me in rank and promotions and time in service. When we would go out downtown and he would get the beer in him, it would start. I out rank you and you have to do what I say. The fight would ensue and he would lose every time. Well it came time and he made Sergeant. When he pinned on his stripes, I told him respectfully that if he still came out with us and started with the I out rank you stuff, he was going to get whipped like a Private because that wasn't how an NCO acted. I also told that to the Detachment Sergeant. Well we went out one night and he decided to go with us. Sure as fire, he started his I out rank you stuff. I asked him twice to stop to no avail. The fight was on and he lost that night so severe they had to pull me off of him. He went to the Detachment Sergeant to bring me up on charges and the Det. Sgt. told him "If you are gong to act like a private I can make you a private". So even as an NCO, you still have to give respect to get respect.