My parakeet nest is too small. Can I move chicks? Will parents care?

As long as the parents are caring for the babies, I would leave them to do their own thing. I wouldn't move or touch the nest while the babies are still young. Once they're a bit older and moving about, you can then clean the nest, or clean it in between clutches. Having baby bird food and a syringe on hand is always a good idea in case you need to step in, notice they aren't thriving or growing well, or the parents aren't caring for them.

You can try to move the eggs if the nest is beyond repair, however do note that the eggs are EXTREMELY fragile, and are likely to break or crack even with gentle handling. If the nest is still viable enough that she is able to properly lay on the eggs without her or the eggs/hatchling being harmed, then not disturbing it, or disturbing it as little as possible is best.

You can clean out the nesting box as needed. Gently scoop the chicks out and place them in a clean temporary box. Expect mom to complain and peck. Scrape the box and fill with clean nesting material. Return the chicks to the box afterwards. If the hen is not feeding them adequately, you will need to supplement​ them or pull them for handraising. Use a commercial handfeeding formula. Follow label directions. If you don't have experience hand feeding, ask your avian veterinarian or an experienced breeder. If this is Boo Boo's first clutch, the loss of some of the chicks may be due to inexperience.

Keep the nest box clean. You can gently scoop the chicks up in a paper towel, clean the box, add clean nesting material and return the chicks. Make sure Noshi is eating a healthy. Ideally, she should be eating a commercial pelleted diet such as Exact or Pretty Bird. You can supplement with a fortified seed mix and fresh fruits and vegetables. Monitor the chicks for full crops (which means they are being fed) and weight gain. If necessary, a commercial hand feeding formula can be used to supplement the chicks.

It may be that the egg was actually pushed out of the box as it is not viable. If they will not let you in the box (where there may be other eggs), or the egg is not viable, you can simply remove it. Otherwise, waiting until night when both birds are calmer and quiet may give you a chance to place the egg into the nest.

If you are hoping to raise babies, then providing nesting material for the parents is best so that they can make a safe nest where the eggs are less likely to fall or break. The parents may also break eggs or discard them if they aren't viable or they are young parents that don't know what to do. Because the eggs need to be incubated, they need to remain in the nest with the parents in order to hatch. You can keep letting both parents try to raise a clutch (and be sure to provide a calcium supplement for the mother), or, if you don't wish to raise babies, can just remove and discard any laid eggs.

Momma bird may be a bit stressed out now that there are new babies to care for. Adding in extra food and water dishes so that each bird has one to eat at without fighting, and making sure both birds have their own perches to relax on can help reduce tension. As for the babies, momma bird should care for them herself, however having some commercial baby bird food (most pet stores will carry it) can be useful in case she can't care for them all on her own. As they get a bit older and start to explore, you can start to offer treats and pets to get them used to being handled.