How closely does an animal have to be related for hybridization to be possible?

 found this site showing several hybrid birds .  No parakeet-pionus hybrids shown maybe because parakeets are from Australia and pionus birds live in S America .

I assume you are talking about hybrids of two different species.
Dogs and wolves are the same species, different subspecies, and they interbreed freely.  Coyotes are classed as a different species, yet they interbreed freely with wolves and with dogs, so I wonder whether they ought to be in the species lupus after all.  
There is not an easy way to tell.  I think most likely it has to do with the two species having the same number of chromosomes, so that a half-strand of one species' DNA will be able to zip up with the half strand of the other species' DNA.
Two species that can hybridize usually come from the same genus, such as horse, donkey, and zebra, which are all in the Equus genus.  Or lion, tiger, and leopard, which are all in the Panthera genus.
However, I discovered one case where the two species are not in the same genus:  Rock pigeon and ring-necked dove.  These can hybridize even though the pigeon is genus Columba and the ring-neck is genus Streptopelia. They are both in the Family Columbidae - doves - though.  
These two species of doves are both feral and abundant in southern California where I live.  I doubt they would mate in the wild because they would seek out a mates the same size.  But I have had them breed in captivity.  I have also seen two or three of these hybrids in the wild; maybe they were escaped pets, or maybe they did mate in the wild.  But it certainly doesn't happen often or there would be more of them.  They do seem to be sterile, as none of my hybrids produced offspring.

That varies. The usual rule of thumb is that they have to be of the same species. However, there are fishes that are completely interfertile across generic lines. As to your specific problem, no way.