I would not try to catch the birds yours self. They seem as though they could be already adapted to their environment and believe me they can survive the NY climate. There is a group of Monk Parakeets or Quakers that live in North East New Jersey - I think Edgewater or that area. They have a built a successful colony that has been there for about 5 years. There was a big argument because Quakers are illegal in NJ due to agricultural protection. Fortunately since there isn't any agriculture in that area the people of the town fought to keep them there.
Remember that birds are not as domestic as dogs and cats. Left alone to their environment and each other they will automatically revert back to being wild birds and can be very dangerous (anyone that has been bitten by a parrot can attest to the pain)
Someone probably didn't clip their wings or let them go because they didn't want them and if they were breeders than they are one up already towards surviving. Definitely call your
local animal control department.