More than 120 MPs have written to the government asking for an inquiry into how family courts in England and Wales treat victims of domestic violence.
At least four children have been killed by a parent in the past five years after a family court granted access.
Dozens of parents have told the Victoria Derbyshire programme their abusive ex-partners were granted unsupervised contact with their child.
The Ministry of Justice said a child's welfare was always the priority.
"Where there is evidence of domestic abuse, the courts are bound by law to consider potential harm to the child and this overrides any presumption of parental involvement," an MoJ spokesman said.
When parents separate and cannot agree arrangements for their children, a family court judge can make a legally-binding decision on contact - including whether visits to a mother or father should be supervised.
The fundamental presumption in law is that it is in the best interests of the child to have contact with both parents.
But it has led to the courts ordering children to have contact with an alleged or known violent ex-partner - including some convicted of rape, assault and drug offences, the BBC has learned from dozens of affected parents.
Due to legal restrictions the BBC cannot always view family court documents to substantiate the claims made by parents.
However, analysis of serious case reviews for England since 2014, shows four children have been killed during access granted by the family courts.
It indicates that four further children had been sexually abused or seriously injured, or both.
In all of the cases, social services had been aware of a history of domestic abuse allegations against the partner. The youngest victim was five months old.
In one incident, a father who went on to kill both of his children was granted unsupervised contact with them in an interim order. The case review found that numerous professionals - including a social worker, a teacher and an official from court support service Cafcass - had been too scared to be left alone with him because of his aggressive behaviour.
Other cases refer to a parent having what is known as a "toxic trio" of addiction, mental illness and a history of violence.
The number of children injured by a parent during court-ordered contact may in reality be much higher, as serious case reviews are generally conducted only in cases of serious harm or death.
They are carried out in order to find ways that local agencies can improve their safeguarding practices.
They do not investigate the family courts, so it is not possible to establish whether the courts' decisions led directly to the deaths.
"I was completely naive about the family courts," says Mary - not her real name.
She has spent £130,000 in legal fees to try to protect her children.
Her ex-partner has been awarded regular, unsupervised overnight contact with them.
She says they have since come home with injuries - and she has taken them to A&E.
Mary says her ex had previously been abusive towards her - which began when she was pregnant, "pushing, slapping, throwing me across the room".
After they separated, she called police when he turned up at her home uninvited and was aggressive towards her.
Mary says she has taken her children to hospital after they came home with injuries
He later applied to the family court for unsupervised contact with the children, which was granted.
He had numerous criminal convictions for violent and drug offences.
"I assumed they'd see that to enable a violent man to have a relationship with his children, contact needed to be supervised," she says, of the family courts.
"But that's not how they see it at all.
"My solicitor told me, 'Unless he's beaten you black and blue, he'll be deemed a good enough father. Don't even bother trying.'"
Mary says her children now "wake up sobbing, and I just reassure them that no-one is expecting them to stand up to their dad".
She adds: "Nobody is saying that a child shouldn't have a relationship with their father. It just it needs to be healthy and safe."
The BBC has learned that 123 MPs from seven different parties have now come together to sign a letter to Justice Secretary David Gauke calling for an independent inquiry into the family courts "to establish the extent of the problem and if more fundamental reform is required".
The letter continues: "The lack of transparency in the family courts, while essential in maintaining the privacy of families and children, does not allow scrutiny and masks decisions that are made contrary to the interests of victims of domestic abuse, rape and violence - or their children".
Louise Haigh says it is "horrifying" that men are being granted access to their child after being convicted of a violent crime
Labour's shadow policing minister, Louise Haigh MP, said it was "horrifying that even in proven cases of sexual assault, severe domestic abuse, rape, murder in some cases, men are still being encouraged and granted access to their child".
She added: "If they're a known risk to mother or child, then we need to assume that contact probably isn't best for the child and grant it only in certain circumstances."
When there is a court-ordered contact, a parent can be at risk of being fined or going to prison if they fail to send their child on the unsupervised visit.
Barrister Charlotte Proudman, who specialises in cases involving violence against women, told the BBC she had witnessed a perception that mothers were preventing contact with fathers without good reason.
"I've heard judges say, 'Oh, it's just a little bit of domestic violence.' It's minimised rather than seeing the significance of that," she added.
The Children and Family Court Advisory and Support Service (Cafcass) said in a statement: "One of our most challenging professional tasks is to assess what level of parental involvement is safe and in the child's best interests, in cases where a parent has a history of domestic abuse.
"We must continue to reduce [the risk of parents harming children] by understanding these cases better and looking wider than the court process."
A spokesperson for the UK judiciary said judges were "required to consider all the evidence put forward and to reconcile any conflicting interests at a time that they know is exceptionally stressful for all those involved."
The Ministry of Justice said: "The welfare of the child is the paramount consideration of the family courts when making decisions about their upbringing".
It added that it would "continue to explore" ways to improve how the justice system deals with domestic abuse.