I wouldn't say I'm an expert (thanks for the compliment, Blu), but here goes...
They're a commercial fruit tree pest in India, so I'd be a bit concerned about your fruit trees. (In the wild they eat fruit, berries, seeds and flowers; here they also visit nut and seed feeders and also take meat scraps.) My mother used to put thread round our cherry tree to keep birds off. Perhaps some similar trick might work for apple trees. I've noticed fruit trees with CDs hanging from them - I presume that's to deter birds, but I don't know whether it's effective.
Whether they're a threat to native birds is not altogether clear. Competition for nest holes is thought to be more of an issue than food, as most of our native garden birds (e.g. tits, robins, blackbirds, thrushes) mainly eat small invertebrates, particularly in the breeding season. The tits should be OK as the parakeets would need much bigger holes. The larger woodpeckers (Great Spotted and Green) are pretty tough birds and would probably be able to see off the parakeets. (The Lesser Spotted Woodpecker's holes would be too small). Starlings would seem the most likely bird to suffer from nest hole competition - their numbers have declined around 70% since the 1970s for reasons unrelated to parakeets.
There are some wonderful urban myths about how they came here. CountessDracula has already mentioned The African Queen. There's also a story that they're all descended from birds released by Jimi Hendrix. I think girlfrommars is broadly correct over the escaped cage birds, but "released illegally" may in many cases have been closer to the truth than "escaped". They are very demanding pets. Here is a website that gives some insight into why owners might be tempted to release their birds - imagine having a shrill green flying toddler in your house for 15 or so years!
To summarise, your apples may be at risk but your garden birds are probably not.