This is probably going to get a bit long, so I apologize in advance. We all have different circumstances and different experiences, so I am real reluctant to tell you what to do. Maybe by trying to help you understand what the issues are, you can come up with the strategy that is best for you.
Broody hens often wean their chicks as early as 4 to 5 weeks of age. These chicks are fully integrated with the flock and usually get along. You don't have to wait until they are grown, but they will still have issues. For some people, especially if space is tight, it might be better to wait.
The first thing to worry about is pure integration. Chickens are territorial and will often defend their territory from any chickens of any size that they don't think belongs there. You'd think this would be the rooster's job, and sometimes it is, but with mine it has always been hens. This is where housing them side by side for a while is really good. After a week or two, the other chickens accept that these new chickens have a right to be there. You may still have a chicken, usually a hen, that seeks out to destroy. Some chickens are just brutes, but this is usually not a problem if they have been housed side by side for a while. Sometimes this does not happen at all and the old flock immediately accepts the new chickens. Sometimes this part goes so smoothly you wonder what the concern was, but sometimes it can get really bad.
Even after the new chickens are accepted into the flock, you still have the pecking order issues. A more mature chicken is always higher up in the pecking order than an immature chicken. If a lower ranking chicken invades the personal space of a higher ranking chicken, the higher chicken is perfectly within its rights to peck the other to enforce its rights. Usually, intimidation works and the lower ranked one runs away. All is again well in the world of chickens. But if the lower ranked does not run away, this is considered a challenge and the higher ranked can get pretty violent while enforcing her rights. This is usually a hen or a non-dominant rooster. This is where room to get away and places to hide become important. You can have chickens that seek and destroy in the pecking order business, but usually intimidation is all it takes.
A quick story to demonstrate. I've seen two week old chicks being raised by a broody eat side by side with adults. Sometimes the adults ignore the chicks, but after a while, one will usually peck a chick to remind them it is bad manners to eat with your betters. The chick runs back to Mama as fast as little legs and wings can get them. Mama generally ignores this behavior. That chick needed to be taught a lesson on flock etiquette. But if that hen tries to chase the chick to do damage, Mama gets enraged and teaches that hen a lesson. Sometimes it takes a little violence to maintain peace in a flock.
I think what you are planning is great. Will a hen try to get through the netting? Maybe, maybe not. A hen may peck through it if a baby chick gets too close, but hopefully it can get away. You might have one of those seek and destroy hens. I don't know. So build it strong. Also, be careful how you build it. I had a chick get tangled up in loose netting one time where it could not get away. It died. I tore out the netting and put in chicken wire so a chick could not get tangled up. Your parents do have a legitimate concern in that a hen can try to get to the chicks, but if you build it right, you should be OK. I keep my brooder in the coop with the older chickens from Day 1. You can do it safely.
A lot of other people have given you good advice on what you can do to help reduce your risk when you mix the two flocks. Provide separate eating and drinking places to reduce potential conflict areas. Give the chicks a way to get away from the older chickens. This may be perches or things to hide under or behind. Mine free range, so the chicks find a place well away from the adults to hang out.
For mine, one area of high conflict is on the roosts, once they start roosting. Sometimes, but not always, one or two hens take it upon themselves to make life miserable for the lower ranked chickens on the roosts. It has been so bad, both with broody and brooder raised chicks, that the chicks start looking for safer places to sleep, maybe in the nest boxes or even outside the coop. I put up extra roosts, higher than the nest boxes but lower and away from the main roosts, to give them a place to go until they get mature enough to make their way on the main roosts.
We are all going to have different strategies because we have different circumstances and experiences. Hope you can find the right strategy for you.
Editted to add. A separate coop, especially linked to the same run or set up with its own run that can be opened ot the main run, is a great asset to have when integrating chickens. You can safely house them side by side in the run and they have a safe place to sleep after you start the integration.
