Rayner deflected the question with a joke about Anderson's own career.
"He talks about CVs, it's good to see [Anderson] doing well on his - once a Labour councillor then a Tory MP and now Reform chief whip," she said.
"Every time he switches party, he gets a promotion."
A few minutes later, Conservative Graham Stuart said all MPs "recognise the authenticity" of Rayner's back story including her time working as a carer.
Stuart then referenced recent cases where people had got into trouble for lying about their experience and achievements.
In June, Nick Adderly, the chief constable of Northamptonshire Police was sacked for exaggerating his naval rank and length of service.
And last month, a woman was jailed for five years after lying about her experience and qualifications to get a job as a senior nurse in a unit for sick and premature babies.
Stuart asked if Rayner agreed that "serious consequences are right and just in such cases".
"I know where he is trying to go with this," replied Rayner before going on to defend Reeves.
"Our chancellor in the last four months has shown more competence than the last four chancellors he had," she said.