Parakeet do not need a mate to lay an egg. The difference is that the egg laid by a single female bird will not be fertile, whereas a female that is part of a pair can lay a fertile egg. A simple way of understanding why a single female produces eggs without a male present is to compare the situation to humans. Each month, women go through a menstrual cycle but don't get pregnant unless, of course, they have been fertilized by a male. The same applies to birds, which can cycle without producing a fertile egg. A cycle will usually result in an unfertile egg.
Yes! My neighbor has about 20! They will lay eggs but the eggs will not result in baby birds! Their must be a mate to fertilize the egg! I learned this recently when her bird laid an egg in a cage by itself!