Labradoodles are in fact a great mix - poodles are like labradors, retrievers - their curly coat makes them very good to retrieve waterfowl. They are a popular mix also because like poodles, they are considered as potentially less allergenic, since they shed less. But most of them do not need clipping, like poodles do. Please don't forget that the poodle is considered an extremely clever breed and is originally a working breed. The fact that even standard size poodles to-day are shown in the silly show clip (even though this originated in the traditional "waterdog" clip, with fur left on the front and loins, and aound the leg joints, for warmth and protection when swimming) all teased out and bow-tied, does not take away the breed's delightful cleverness and friendliness (no, I do not have a poodle - but I would not mind one as a family pet - would in fact prefer them to a labrador, given that breed's tendency to become overweight so easily and their rather phlegamatic temperament).
That said, to come back to the correspondent's question: it is one thing to take a pup to a shelter (at least it seems to be a no-kill one) but how do your parents get a new puppy? Poin its, a good breeder wants to know to what kind of a home their pups go; and will often offer to take a pup back if things don't work out, for re-homing. Good shelters want prospective adopters to answer questions too. Since your parents, if what you say is true, are not really a good proposal for adoption, they would therefore need to take some liberties with the truth. Not sure what you can do about it. Likewise, not sure about what actually is the matter here, from your first post. Are you implying that if you had not had a disagreement with your parents, they would have kept the pup? Why take it out on you? Do you live at home? If so, you ought to share the responsibility of looking after the pup so as to give your parents no excuse. Regardsless, it's not about you, it's about the pup.