Can lethal diseases be transfered from pigeons to parakeets if they are in close proximity?

Actually, it can go both ways.  I actually lost about ten pigeons once to a Salmonella infection that was spread from parakeets that got into my loft (the vets verified that the Salmonella strain was a psittacine one)

However, if the birds in question are both domestics and both are well managed there is no problem. My buddy raised both budgies and pigeons in his back yard for over 30 years and both were hale and hearty.

If you're talking about feral pigeons that are getting into your budgies aviary, then I'd be a bit more concerned simply because all wild birds - pigeons, sparrows, finches, and, yes, even wild budgies can be a source of parasites, etc.  I'd also fix my aviaries so that other birds can't get in.

If, however, you raise budgies and your neighbor raises pigeons - I wouldn't worry about it so long as you're both taking care of your birds.  One concern for both species could be an infection of Chlamydia psittaci  (ornithosis), which is a reportable disease, at least in the States, and one which can be cured.Yes.Yes they can.  If you have wild pigeons, or pigeons from a shady place keep them seperate from your parakeets, and wash your hands and change your clothes anytime you interact with them.