The human rights group is launching a public awareness campaign for the Act, which is said to be under threat from Brexit due to the UK possibly axeing EU laws after the country leaves Europe.
The ads were set to begin appearing at UK train stations today.
Barry Devonside, who lost his son Christopher at Hillsborough, said: “It’s a very small-minded decision for Network Rail to take. Human rights aren’t political; they cover everyone right across society.
A spokesman for Network Rail said: “We first saw the adverts yesterday and on inspection, they don’t meet our rules on ‘political’ advertising.
"We aim to be fair and even-handed when deciding which adverts to accept in our stations and have a set of rules in place to ensure all advertising is treated equally. “The advertisement directs people to sign an online petition, and it is for this reason we deemed it political and against our rules.”
The ads were not deemed too political for the London Underground network, where they appeared as planned from this morning.
Kate Allen, Amnesty’s UK Director, said: “We simply don’t accept that basic human rights are ever a political issue. “The irony is that human rights cross party-political lines and so do failures, cover-ups and abuses.
"Successive governments managed to let down the Hillsborough families, and their story demonstrates precisely why we need enduring human rights protections which aren’t redrafted by the government of the day, on a political whim. “It’s vital to safeguard basic rights in these politically volatile times as constant protections relied on by ordinary people.”