Despite the fact we are selling, we want to rent in Crete. It would obviously be best to take the advice given elsewhere and book short term accommodation initially and look for a longer term place once we are there, and see what we are getting.
That is VERY smart!
As for long term rental: you really should look by yourself and not through an agent. This is probably cheaper and will give you much more choice as agents are not the usual route people go when looking for a place to rent or trying to rent a place.
Along those lines, I don't see you being able to arrange a long term rental a long time in advance because most of the places that you'll find for rent are for rent now, not in a few months time.
Would it not be simpler to arrange for a short term rental that can take your pet parrot? Alternatively someone on this board might want to parrot sit?