Mr Gallagher points towards The Principle in NHS Car Park Charging rules which says car park charges must be reasonable.
He then claims it cannot be considered reasonable to charge the NHS workforce to park at work during the crisis or reasonable to distract the workforce with concerns over being fined.
He called on Health Secretary Matt Hancock, Chancellor Rishi Sunak and Prime Minister Boris Johnson to take action in abolishing the fees.
In his petition statement, Mr Gallagher said: “Mr. Hancock, Mr. Sunak, Mr. Johnson, if you really are as grateful as you say you are to the staff of our NHS, then please abolish staff parking fees and fines throughout the NHS immediately and forever.