15 month old "holding it"

Hi, I am new here and I am wondering if anyone has any ideas. The 15 month old boy started at my daycare a few weeks ago and he tries really hard not to poop. He squeezes his legs together and goes all red in the face. He does it at home too. I think it has started as constipation but he has been given glycerin for a month now by his parents and it does loosen the poo. So when he finally does let it go, there is so much of it but it is soft and not like balls. And it isn't runny or anything. Its just the fact that he fights himself so hard to not poop, that we are thinking he will get himself so blocked up it will cause a lot of problems in the future. I was reading on this forum here about other kids and being constipated and the foods that might work but I think for this kid it will have to be some solution for him physically too. Like should I have him sitting on a potty or something? He drinks lots of liquid. I change him 3-4x/day in 8 hours with FULL diapers. And we eat fiber foods here with whole grains and veggies too. He is new at my daycare so I don't know them well but I think the parents are on board with whatever, too. He has been to a couple of doctors already.
Thanks in advance. He is a sweet little boy and has fit right into the daycare and I am worried about him because after a few days of not going he isn't himself and he has a bad day.

You say drinks lots of liquid but what is drinking? I'd be giving him water only with milk maybe 1-2x a day (including at home )

I'd be making some high fiber pancakes, muffins, bread etc.

Does it hurt for him? He's quite young but is it because he doesn't like the feeling of poop in his diaper and maybe time to toilet train? I've had one child well under 2 who was fully trained and a friends aunt is famous for her 3 children being trained under 2 so it is possible lol, just rare I think!

I had one like that. Every time he went crossed legged and red face I rushed him to the potty and sat him down. He would always then go. I think it helped when he sat down and I spread his legs to,the side to,open his bum to make it easier to poop. Lots lots lots of water will help to.

Haven't had this experience yet. But if it were me, I would try to explain that the food goes through the body and needs to come out or else our tummy will hurt. Not sure if you can find a book about it and read it to him. Maybe he's really uncomfortable with the feeling of poop in his diaper...it might be a sign that he's ready for the potty. Maybe next time you see him holding it in bring him to the potty and try on there. Hope that helps!