Emergency services were called to the scene in Dalgety Bay, Fife, on Saturday after the boys were found. They were taken to Victoria Hospital in Kirkcaldy where they were pronounced dead.
The twins' parents 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. 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 the family only recently moved into the rented home and the previous resident, Keith Henderson, ran a koi supply business from the property that required the installation of a fish tank at the side of the property.
The ornamental fish are collected by enthusiasts and tend to require large, deep ponds.
Devastated friends placed flowers outside the family home in a quiet cul-de-sac.
One said: "Rest in peace little ones. Hold each other's hands and stay close."
Another said: "Rest in peace lovely ones."
Keith McKinlay, 50, a Gulf War veteran who used to live beside the couple said: "Merv worked at the industrial estate in Dalgety Bay and so did Sarah. They are just laid back and a lovely couple and so easy to talk to. They were easy going and pleasant and did so much for me.
"I'm still in shock. You can't put that into words. Sarah must be devastated because she is quite a sensitive person. It must be devastating for the both of them.
"I know she wanted children and that she was chuffed when she found out they were expecting twins. Why take it away from her? It's just so unfair.
"I know they were trying hard to get the kids. I know she wanted kids but that there was some problem."
He added: "They will both be devastated. It's so unfair. It should not happen to anybody. My heart goes out to them."
Officers are supporting the family and working to establish the full circumstances of the incident.