At Philmont is it permitted for a father and son to share a tent? On our crew, logistically this would work better.

No.
Never.
Boys lead, work, sleep together. Adults stay in the background and sleep there too.

Mark,

You ask if it is permitted. The answer is yes, but it is not recommended.

When I went to Philmont in 2009, I was the crew leader. I also slept in a tent with my father. This did not affect the crew dynamics, as I was still the youth leader, and got advice and counsel from the adults.

That said, that situation may not work for every crew. Sometimes the boy that sleeps with his dad could be considered an outsider.

Allowed? ... yes. A good idea ... not so much. The other responses have made a good start at offering reasons why. Short version, it can be an obstacle to youth crew cohesion.

I was in a 2010 crew, 5 youth, 3 adults and chose to carry a bit of extra weight so that a dad and son wouldn't have to tent together. That crew became a real tight unit. The other crew in our contingent had a similar situation and had a dad and scout tent together. It ended up reinforcing a pre-existing sense of separateness on the part of the youth in question, much to the sadness of the dad.

I think it depends on your crew. We had 13, two lady adults, another male adult, and me, (adult most times, lol). To make the "best" use of equipment, we had the other male adult sleep with his son, the ladys' slept together, and I slept with my son and his best friend, who I have known since he was 3. we had 3 three man tents, and two two person tents = 13 exactly. Our crew was from one troop, and the boys knew each other extremely well, it made no difference who slept where. YMMV, use your best judgement, it is allowed.

Yes, it is allowed.
Our ranger actually suggested it to minimize the number of tents on our trek.

Usually a bad idea, but I can see where it wouldn't hurt on occasion.

I would not do it and I try hard not to be a "Parent" to my sons at Scouting events where my boys call me Mr. Hanson instead of Dad. I would not deprive my sons of a quality Scouting experience just because logistics and safety require adults to participate in Scouting and that means they are stuck with me going on most outings. Ok... so I haven't bothered with a tent in a long time, but the principle is still the same even if we are all sharing one of God's beautiful blankets of stars.

As other posters have noted, it is permitted and sometimes encouraged. Philmont does not want any scout in a solo tent and appears to tolerate adults in solo tents.

Our crew of 12 had two dads who shared tents with their sons. I and one other dad both slept in solo tarp tents.

I do not think the sleeping arrangements impacted crew cohesion because we had done multiple backpacking trips as a troop and three specific hikes as a crew to prepare for Philmont.

Each scout and troop is different, but I was the adult leader and my son preferred to act as though we were not related except for about 45 minutes a day during the lunch break.

BSA Youth Protection rules do prohibit any adult from sharing a tent with a youth that is not his or her son.