The children were found at the house alongside their mother Terri Harris and Lacey’s friend Connie Gent, 11.
Mr Bennet reposted a picture on Facebook from August last year by Tom Bradley, from California, in the early hours of Tuesday morning calling to bring back capital punishment.
The message, inside a rainbow heart, said: “Bring back the death penalty anybody else have enough balls to post.”
The force said a 31-year-old man has been arrested on suspicion of murder and remains in police custody.
Mr Bennett also asked supporters to wish his “rock” partner Caroline Harrison a happy birthday.
He added: “John and Lacey were looking forward to it and it breaks my heart we cannot celebrate as one big happy family.”
He posted an emotional video calling his son John Paul “My mini-me”.
He wrote: “My heart and soul of my life. You brought me laughter every day.
“I wish I could have been there and saved you to stop all the pain suffered and told you everything will be ok. But now I've suffered the worst thing any parent could go through.”
It came after Charlie Gent paid tribute to his daughter Connie, saying: “Parents, spend as much time with your kids as you possibly can.
“If you’re a parent and arguing please don’t stop that child from seeing the other parent. You never know what’s round the corner.
“I’ve learnt the hard way like most things in life and it’s too late and I don’t want anyone to ever go through what I am going through right now.
“My little girl has gone, I can’t even put into words or even how to feel or react right now. I shouldn’t even be writing this, now I have to live with the fact I was helpless to protect her as I always promised and she’s gone.”
He added in the social media post: “Fly safe baby girl I love you with all my heart and always will. Going to miss you like crazy.”
Many people were in tears as they stood and listened to prayers from Killamarsh street church pastor David Taylor.
In a statement released on Monday afternoon, Chief Constable Rachel Swann said: “This has been an understandably shocking event that has deeply affected the local community in Killamarsh, as well as Connie’s family and friends in Sheffield, and our thoughts are also with them.
“I would also like to state my thanks to those officers who arrived at the scene with such speed as well as colleagues from the East Midlands Ambulance Service who attended the incident.
“All acting with immense professionalism in such a difficult situation.
“While understandably worrying and upsetting this is believed to be an isolated incident in which those involved were known to one another and nobody else is being sought in connection with the deaths.
“The force did not have any prior contact with those involved.”