We will start with the tank, your tank has a base I think of 15"x15". Traditionally breeding tanks are made long, They are Twice as long as they are high and deep ( eg 24"x12"x12" ) The reason for this is to give the fish room to run. I would never put Corydoras into something smaller than this tank 15 gallons. The shape is important. You have managed to get these guys to spawn, great and we can work with what you have got. Because of the shape and the fact that everybody is on top of each other, the parents have to come out. The baby guppies can stay there they are fine.
I would normally move gavel away from one end or the front of the tank by about 3 inches this gives a section of bare bottom. Important for cleaning and observation. Any food you put into the tank is a suspension it will fall to the base of the tank and will end up at the lowest point, the bare base of the tank. The rest of the base doesn't need to be disturbed, just syphon from the bare base you will see material draft down as you syphon. Only do your normal water changes, make sure you rinse the sponge filters, don't have the filters running to hard.
Feeding egg yoke has been described elsewhere. But use some cotton an old singlet or any cotton you have experiment with different material. The idea is that for new fry the particle size of the egg yoke should be so small you can't see it with the naked eye. The suspension in the tank should look like milk. It only takes a tiny bit of egg to achieve this. That suspension will dissipate into the tank the young fish will pick up those tiny particles. I only ever feed twice a day. Morning and night.
I have said it is possible to leave the parents in the tank. I have done this with a tank 48"x18"x18'' The tank was heavily planted at one end. Once the fish stopped spawning you feed them. The next morning they spawn again and you feed them once the fry hatch they migrate into the gravel bed where they are largely safe, feed egg yoke which drifts to all parts of the tank the fry get food. I have explained why egg yoke feed this way is so good, remember this is first food, I would feed this for one to two weeks depending on the setup and number of fish. Ask more questions if you like or don't understand what I have written.
Yeah, I didn't plan to breed the cories in this tank, it's a temporary tank I got to use as a quarantine tank, and as a grow out tank for guppy fry. The cories are to go in a 57 gallon, but they spawned in this tank before I moved them, lol. I wouldn't keep them in this size tank permanently, bronzes get too big to live in a smaller tank long term.
My other tanks are heavily planted, but since this one was only meant to be used as a quarantine and grow out, heavy planting isn't great for catching fish or quarantining, and the tank light isn't really good enough for plants to flourish.
I think if a breeding tank was designed, the right shape and size etc, that raising some fish with the parents would be fine, and perhaps even preferable. Not sure it would work with cories though, as
@mbsqw1d has experienced, cories aren't voracious predators, but they're naturally dopey and scavengers, I could see them eating fry without even noticing what they're eating! I would imagine a lot of fry would get eaten, even in a heavily planted tank with the adults present, but the odd fry surviving.
I'm not opposed to trying some egg yolk, a lot of people do use it. I don't think I'd want to feed it exclusively for the first two weeks though, when the first weeks of development are so crucial for any creature. Would egg yolk provide all of the essential nutrition? Like when feeding adults, I prefer a variety, and would want to feed some live food too. First Bites provide a variety of nutrients, Liquifry also provides a dense liquid food for tiny fry, and I'll be culturing microworms too, which I've seen recommended for cory fry. Also the micro organisms produced by almond leaves, alder cones and moss.
I wouldn't recommend gravel with cories though, adults or fry. Their barbels are too easily damaged, gravel harbours too much dirt and bacteria, and doesn't allow them to practice their natural feeding behaviours. Sand is always preferable for corydoras.