[Monica]

Parent birds do indeed feed the chicks throughout the night as well as during the day until the chicks are old enough to go without feedings during the night.

Here's some links that may help...
J-Birds Hand Feeding Page
Practical Avian Pediatrics
Parrotlets: Introduction to the Care and Breeding

Tiel info but good for a reference.... can't seem to find much on handfeeding parrotlets
Winged Wisdom Pet Bird Magazine - The Responsible Breeding of Cockatiels Part III - All About Handfeeding Cockatiel Chicks

This is for tiels, but gives you an idea...
Handfeeding Step-By-Step

BROODERS:
Now that you have decided which hand feeding formula and utensils you plan to use, it is time to choose the brooder. A lot of your decision on what to use for a brooder should be based on the age of the babies you are pulling, and how many babies there will be. A small amount of very young babies will do best in a dark brooder at about 95 degrees Fahrenheit, while a larger amount of older babies will do better in a glass fish tank at a lesser temperature. Whatever brooder you use should first be cleaned and disinfectant, making sure you rinse the brooder thoroughly to remove any disinfectant residue. After the brooder is clean and dry, I put about a couple of inches of pine shavings on the bottom of the tank and cover that with a thick layer of paper towels. The shavings will help to distribute the heat evenly over the bottom of the tank, while the paper towels will help to absorb the moisture from the chicks' droppings. I then place a heating pad under the tank, and cover the tank with a plastic tank cover or wire grill and a large bath towel, leaving one corner open for air circulation into the tank. Be sure to heat the tank up to the correct temperature before you add the chicks. A non-breakable thermometer at "chick level" in the tank along with this temperature guide should help:

SCHEDULE:
I usually pull my babies at 2 to 3 weeks of age. I have handfed chicks as young as 1 day old but, only if there are problems that require me to intervene at that early age. If you wait much longer than three weeks it can be difficult for your chicks to accept handfeeding; however, I have been more successful at an older age with a bent spoon than with any other utensil. If you pull the chicks earlier than 2 weeks, they may not get all the natural immunities they need from their parents; at that point, it is important to add an avian specific lactobacillus acidophilus (a "good" bacteria) to the formula if it doesn't already contain this.

I will not start hand feeding until the newly pulled babies' crops are empty. I prepare the formula according to the directions on the container, and use a candy thermometer (available at most supermarkets) to make sure the formula is 102-104 degrees Fahrenheit before I start feeding. If using a syringe for feeding, begin by placing the baby on the table in front of you and inserting the tip of the syringe gently into the left side of the chick's mouth. (IMPORTANT: when the chick is facing you, his left is YOUR right.) Point the tip toward the right side of the chicks mouth. Carefully with a slow, even pressure on the syringe plunger, begin dripping the formula into the chick's mouth. You want the formula to be fed slowly so that the baby recognizes you are feeding it, and so the formula goes down into the chicks crop and not the windpipe. The amount of formula depends on the age and weight of the baby. If you keep an eye on the chick's crop, you will see it filling as you feed. Make sure you do not fill the crop to the point where the food comes up into the chick's neck; it should also not appear pendulous or over-expanded. It is a good idea to try a little at a time until you determine what is enough. Here is the handfeeding schedule that I use:

You'll note that the amount per feeding gradually increases as the baby gets older, until the time of fledging. During this time, you'll be hard-pressed to get the baby to take much food. Do not let this fool you into believing the baby is weaned. Instead, it is "dieting" to remove the large abdomen they have as babies so that flight will be attainable. You'll note that, during fledging, babies seem more interested in flying than eating. You must make sure you hold them down during feeding, as they will take off clumsily, and may injure themselves upon landing. A chick may lose 10-15% of its peak weight during fledging.

As the chicks begin to feather out and need less heat, I place them in a small, grateless cage with low perches so the chicks can reach them easily. On the bottom, I place a wide variety of foods, pellets, and seeds. Once they are able to perch, I add a dish of fresh water, I place the food in dishes, and add a grate to the bottom. As the chicks get older I add more perches at higher positions to encourage them to fly and exercise their wings. It is very important to have a gram scale that has readings in one gram increments to check your chicks weight during the hand feeding and weaning process. If you notice the babies not gaining weight (or losing too much weight during fledging) you may need to offer formula more often, or it may be indicative of a problem which needs a vet's attention, such as Candida (yeast).

The Cockatiel Cabin's Handfeeding Procedures

A feeding schedule should be set up and followed as closely as possible. This is especially important for very young chicks. Chicks who are 1-3 days old should be feed every 1 - 1 1/2 hours around the clock. They should be able to go from midnight to 6 a.m. starting with the 4th night. Since the formula is being mixed at a thicker consistency, the digestion takes a little longer and can sustain them through the 6 hour span. We continue to feed the babies on a 3 - 4 hour schedule during the day until around midnight. The morning feeding time varies due to the emptying rate of the crop. If a baby has not digested all of the formula from the night before, we will give him some warm pedyalite or lactated ringer solution to keep the food from becoming too dry and impacting the crop. Once all of the formula is gone from the crop, we feed the baby his breakfast and begin our schedule for the day. If the crop fails to empty, we then begin using the papaya mixture discussed on the Slow Crop Page.