Michael McCelland, whose son Craig was killed in Paisley in July 2017, and Neil Bibby, Labour MSP for West Scotland, said such inquiries should be automatic where tags are involved and families should not have to plead for one.
James Wright stabbed Craig McClelland, 31, a student, from Paisley, five months after removing his tag and breaching his home detention order.
Wright, 25 convicted of Mr McClelland’s murder last year, had been at large for five months at the time of the attack .
Calls for an independent public inquiry have been refused by justice secretary Humza Yousaf. Mr Bibby is also proposing amendments to the Management of Offenders Bill to enshrine their right to an inquiry in law.
The family have also written to the Lord Advocate asking for a fatal accident inquiry.
Mr McClelland said the family should not have to beg for an inquiry.
“As a family we will not have any confidence in the system, until lessons are fully learned.
“So far our calls have been ignored and the justice secretary is blocking a public inquiry for no good reason. We should not have to plead for an inquiry, it should happen automatically.
“If Humza Yousaf and the Lord Advocate can’t see the need for an inquiry in cases like these then the matter should be taken out of their hands.”
Mr Bibby said: “It is critical that confidence is restored in the system and this cannot happen until lessons are learned.
“This is why it is so important a full and independent inquiry takes place into the failures that led to this tragic murder. An inquiry should be automatic.”
A Scottish Government spokeswoman said: “As the justice secretary told parliament in December, having carefully considered the circumstances, a full public inquiry was not the appropriate way forward, given the reviews and actions already taken in response to this case.
“There have been substantial changes to the process for approving prisoners for release on Home Detention Curfew as a result of two independent inspectorate reviews, with the Scottish Government, Police and SPS accepting their recommendations in full.”