A potential problem with cloth is chewing and ingestion of threads. This is a known cause of fatality as the particles lodge inside the digestive system and eventually cause obstruction.
Not all birds will chew and swallow, but the risk is high. Are you able to offset the cloth so it is not within reach of their beaks/feet?
You can hang a shower curtain or fabric attached to ceiling and set back from the cage to offer privacy. Or use a folding room divider. Parrots do t like living in the dark, so make sure the cage still recurves enough light.
More important I believe that tropical birds, especially conures, receive more light than too little. The days are long in equatorial countries, and sunlight absorption is necessary for healthy beak, skin, feathers, eyes, etc. Don't keep them in the dark, but when darkness falls, ensure it's pretty dark so they'll sleep soundly.