Alex Kear, who sits on Wakefield Council as an independent, admitted two charges involving a child under the age of 13 last week.
However, the local authority is powerless to sack him, because of legislation dating back nearly half a century.
Councils can only remove elected members if they are jailed for three months or more.
Kear will be sentenced on August 20, meaning that, unless he resigns beforehand, he can continue claiming his taxpayer-funded allowance until then at least.
Normanton, Pontefract and Castleford MP Yvette Cooper said the seriousness of Kear's crimes means he should not be allowed to continue serving.
She said: "In circumstances like this Wakefield Council should be able to take immediate action.
"Councillors have responsibility for important decisions on the welfare of children.
"If someone is convicted of or pleads guilty to serious sexual offences against children, there should be a clear and proper process to remove them from the council.
"I’m writing urgently to ministers to ask for a review of the law – which dates back to 1972 - to make sure councils can take action and can make clear that children’s safety always come first."
Kear was elected to serve the Airedale and Ferry Fryston ward in 2019, taking the seat from Labour.
Local Democracy Reporting Service
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