The family of Awaab Ishak, who died from a respiratory condition shortly after his second birthday, say they have seen nothing to indicate that his death will serve as a "defining moment".
An inquest this week into the death of Awaab in December 2020 found his respiratory condition developed as a result of mould in the one-bedroom flat in which he lived with his parents, Faisal Abdullah and Aisha Amin.
The flat in Rochdale, Greater Manchester, is owned by Rochdale Boroughwide Housing (RBH), and concerns about the mould had been raised multiple times with no action taken.
Gareth Swarbrick, chief executive of RBH, has refused to resign despite calls for him to stand down following the conclusion of the inquest.
The statement released by the family's lawyers on Friday read: "Senior coroner Joanne Kearsley said that the tragic death of our beautiful Awaab should be a 'defining moment' for the housing sector.
"At present, we see nothing to indicate that the death of our son will, in any way, serve as a defining moment.
"Accountability must be done and be seen to be done.
"RBH have an opportunity to demonstrate to society at large, their residents and not least us as a family that they understand the gravity of this situation.
"We await further action from RBH."
Mr Swarbrick apologised to the parents and the rest of Awaab's family, but said he will not be resigning.
"The conversation around my position has begun to overshadow the most important part of all of this, which is that a family has lost their child," Mr Swarbrick said in a statement on Thursday.
"Having spoken to the board, I can confirm that I will not be resigning. They have given me their full backing and trust to continue to oversee the improvements and changes needed within RBH."
The housing ombudsman is now expediting three investigations into RBH and has asked for a meeting with Mr Swarbrick.