It is fine for the birds to interact as long as they are being supervised. The Green Cheek conures are very gentle as conures go, but he still has a more powerful beak than the parakeet and could accidentally injure it. If they ever start to fight in the future, they need to stop being out together. As far as diseases, both birds have been together so if either had a disease, it would have shown up already. Fungal infections are not that common as long as you feed fresh, clean seeds or pellets. And parakeets are not more likely to get one than other birds. If the birds are on a seed only diet, you should consider changing to pellets or our foraging diets, which are nutritionally balanced the same as a pellet, but contain whole ingredients to offer natural foraging enrichment.
Thank you for asking Lafeber,