Surgery was suspended for two weeks in March last year at Leeds General Infirmary's paediatric heart unit after data suggested a higher death rate than average.
A year-long review has concluded it was safe.
It also said some families of very sick children had received poor care and some parents say they still have questions.
Siobhan Casey's four-year-old daughter, Mylee, died in March 2013 after an operation to remove a build-up of muscle on her heart.
Mrs Casey, from Rossington, near Doncaster, said she was told the surgery was not complicated and was low risk.
"Mylee went down for the surgery. When she came back four hours later they tried to wake her up but she never woke up.
"Two days later I was told she was brain dead.
"I still do not have answers to how or why that happened. I am still waiting for an inquest to be held."
Mrs Casey said she was grateful parents were now being listened to and said the recommendations were a "step forward".
"I think that they are accepting there have been some mistakes and they are doing their best to ensure no other parents go through what we have.
"I am trying to keep an open mind. It is positive and a step forward, but sadly I don't have a lot of faith in the trust at the moment."
Michelle Elliot, whose daughter Jessica was left with brain damage following a stroke as she awaited a heart transplant, said her family were let down by the unit.
Mrs Elliot, from Doncaster, said she was told the 12-year-old was not "sick enough" to be put on the transplant list.
She had a stroke just a few days after being put on the list. She was given a transplant in Newcastle, but remains disabled by the stroke.
"We are living with this every day," Mrs Elliot said.
"She [Jessica] had a dream, a wish, with one of the charities to go to Florida, which has been pending for two and a half years for when she got better.
"When I said 'we can go on this now' she said 'Mum would you want to go in my body. I am not going anywhere' and that really hurts."
Mrs Elliot said she was pleased the trust's director had said he was "profoundly sorry" for what had happened.
"It is a relief. It is a relief to be believed. It is a relief to be hopefully able to move on.
"Until the recommendations [in the report] are in place and we can see that they are working it is only a small step though.
"The mortality report is not a clean bill of health. There is a lot of work to be done," she said.
Richard Graham from Ripon said his family had had a very different experience of the unit.
"When my son was three years old there was a problem with the function of his heart - it was cardiomyopathy, which means his heart was beating too fast and not pumping blood efficiently," he said.
"We were told by medical staff that he was near death's door - those were the consultant's words.
"But he's 10 years old and fine now and I put the fact he did survive down to the professionalism of the team at Leeds.
"When this happened we were in a very vulnerable place and we had nothing but care and compassion from the staff - from the cleaning staff to those on the clinical side.
"For the families that have had bad experiences there are thousands like us who've had very good ones."

"A Little Chaos" follows Kate Winslet's character Sabine, who is chosen to build one of the main gardens at King Louis XIV's new palace at Versailles.
Rickman plays King Louis in the film, which is his second outing as director.
The 11th Glasgow Film Festival will consist of 174 events, including 11 world premieres, and is set to run from 18 February to 1 March.
It will kick off with US writer and director Noah Baumbach's latest film, "While We're Young", starring Ben Stiller and Naomi Watts.
Force Majeure, which won the jury prize at Cannes, will be the closing gala.
Some parents of children treated at a heart surgery unit that was temporarily closed last year have said a report into its safety is a "relief".
[NEXT_CONCEPT]
Acting heavyweight Alan Rickman is to attend the Glasgow Film Festival in support of his latest work.