Absolutely yes, parrots do grieve, and they should be given the same respect as you would a human who has lost a loved one.
Parrots live a very long time, and form strong bonds with their favourite person and other people in the home. They hear and see every thing that goes on, and remember, you are your birds flock now!
Even the death of another animal, in the home that they have got used to having around has an impact on your parrot.
Some parrots will become introverted, not wanting to eat, not wanting to talk, not wanting to play, not wanting to come out of the cage. How long will it last —- many months in some cases.
Some parrots will go almost wild, biting people passing the cage, lunging at people passing the cage, screaming a lot of the time. Tender loving care will get through your parrots grief, but it could take a while.
You will have to be patient and understanding just like you would another human person. Give them lots of loves, give them lots of treats, talk to them when ever you can. Try not to change his routine during this time because this will further confuse him.
Medically there is something that vets can do… they can give prozac which is the drug of choice for birds with emotional problems and it is used often in the States and in England. As a behaviour consultant I can suggest the use of prozac for depression, but I can not prescribe it, that has to be done by the vet, who will know the right doses for different size birds.