Mr Parkin will run an average of 30 miles (48km) a day and is due to pass through Manchester and Sheffield before finishing at Grenfell Tower in London on 20 April.
The avid Liverpool fan, originally from Durham, said he believed a new law “could make a massive change”, citing the long-running fights for justice victims have endured, including those of the Manchester Arena bombings and the infected blood and Post Office Horizon scandals.
Mr Parkin said: “Everybody who’s been affected by this, none of them were doing anything risky at all.
“The Grenfell guys went to bed, the Post Office families went to work and were selling stamps, people went to a football match, people went to a music concert.
“You don’t expect anything like this to happen and your life turns upside down.
“So not only will it reduce that anguish of getting the truth, it also means when you have the truth, you can change.”
Mr Parkin encouraged people to write to their MPs urging them to enact the Hillsborough Law.
A spokesperson for the Hillsborough Law campaign wished Mr Parkin good luck for his run and said it “serves as a timely reminder in an election year of the need to bring in a Hillsborough law”.
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