The cross-party Consumer Protection (Double Charging) Bill will prevent individuals from getting fined or charged again if they can prove they have already paid for the same transaction.
The chair of the Transport Select Committee, Huw Merriman MP, is sponsoring the Bill.
Which? has several ongoing cases where their members are being double charged despite there being no loss suffered by the penalty-giving organisation.
Examples include a man who used the wrong parking area code on a parking app but paid the same amount to the same council as he would have done with the correct area code and yet still received a PCN.
This case was successfully appealed.
Mr Morris said: “The issue of double charging clearly needs addressing through legislation.
"For too long my constituents have been subjected to this sharp practice from receiving a penalty fare for not printing their train ticket despite them having proof of purchase, to being fined for not displaying a parking season ticket on their dashboard. It is simply unfair.
"I very much hope that this Bill will push companies to embrace technology that prevents consumers from being penalised for genuine mistakes.”
Adam French, Which? consumer rights expert, said: “These fines cause unnecessary stress if you face being left out of pocket or having to contest the charges.
"We support measures that could stop you getting charged twice or fined for a service when you can prove that you have already paid for it.”