Rhys and Shaun Scott, who were two years old, were found in the pond on Saturday morning.
They were taken by ambulance from the house in Dalgety Bay to Kirkcaldy's Victoria Hospital where they were pronounced dead.
Police have said inquiries to establish the full circumstances are continuing.
Their parents Mervyn Scott and Sarah Aitken are being supported by specialist police officers.
Both boys were long-awaited little soldiers who were full of love, happiness, fun and cheek."
In a statement issued through Police Scotland, they said: "We would like to thank everyone for their support at this unfortunate and difficult time.
"We have special thanks to all the medical staff, family, friends, neighbours and the police for their efforts, hard work and time spent helping and trying to revive our miracle babies.
"Both boys were long awaited little soldiers who were full of love, happiness, fun and cheek."
The statement added: "Individually and together they have touched the lives of everyone they met.
"We along with our family and friends are still in shock and devastated at such a tragic loss from such a 'freak' accident.
"They have been taken from us far too early and will be forever missed and loved.
"We wish to be left alone and be given the privacy needed to grieve."
It is understood that the family rented the home from a man who formerly ran a business from there, selling ornamental koi carp.
The parents of twin brothers who drowned in a fish pond at the family home in Fife have spoken about the loss of their "miracle babies".