Craig Flowers, who was jailed for a year for supplying cocaine, is performing ceremonies on behalf of a charity chaired by his father.
Humanist weddings were first authorised in 2005 in Scotland.
Independent Humanist Ceremonies (IHC) specialises in non-religious services and has former Scottish Police Federation boss Norrie Flowers as a trustee.
Flowers's son Craig is the public face on the charity's online home page, where he describes himself as a "celebrant and marriage officer".
He is listed as being available for baby naming, funerals and weddings.
However, Craig Flowers has a serious criminal past.
In 2010, he was jailed for one year after pleading guilty to supplying cocaine to his friends while he was a police constable.
He was found with a bag of drugs in 2008 after being put under surveillance by the former Strathclyde force's anti-corruption unit.
After he was caught, a court heard that Flowers told officers: "Look what I got today, it's cocaine. I bought three for £100 to share with my friends. I've got a habit but it's under control - there's no problem."
Sheriff Joanna Johnston said: "Given that you were a serving police officer at the time with a duty to uphold the law, it is particularly concerning that you acted in such a fashion."
According to his IHC biography, Flowers has been working as a "full time" celebrant since 2010 - the year he was jailed.
He adds on the website: "I still get a buzz from helping to make a couple's wedding day perfect.
"I enjoy the challenge of delivering a personalised service that is moulded around the couple's wishes."
IHC states that it works on behalf of "those who seek to live ethical lives on the basis of reason and humanity".
Under the Marriage (Scotland) Act 1977, the Registrar General can authorise celebrants to conduct weddings for a specific time period.
Individuals can also apply for temporary authorisation on a wedding-by-wedding basis.
A celebrant must be deemed to be a "fit and proper person".
After contacting IHC, Norrie Flowers answered the Sunday Herald's questions.
He said his son was self-employed, gets work through IHC and is authorised to conduct ceremonies on a wedding-by-wedding basis.
Asked whether his son informs clients about his past, he said: "He tells people if they ask."
A Scottish Government spokesperson said: "Not being a fit and proper person is one of the reasons why a celebrant who has been nominated by a religious body under section nine of the 1977 Act to solemnise marriages may be removed from the nominated persons' register.
"The same need to ensure that celebrants are fit and proper people applies in practice to any person who has a temporary authorisation under section 12."