The red bus, chosen deliberately to echo the Vote Leave bus with the highly controversial £350m for the NHS claim used in the 2016 referendum campaign, was parked in Cookridge Street on Monday morning.
Crowdfunded by a number of grassroots Pro-EU groups, it has been stopping at various locations around Yorkshire over the past few days.
Members of the Yorkshire Remain Voice Choir serenaded onlookers with satirical songs.
Organisers said the campaign is not party political - instead the main aim is to encourage people to vote tactically to stop Boris Johnson forming a majority government.
Peter Packham, of Leeds for Europe, said: "We've got 10 days left to stay in the EU.
"If Boris Johnson gets a majority in the election on December 12 then we will be leaving the European Union at the end of January.
"Although he says that that's getting Brexit done, it's not going to - we'll be faced with ten or 15 years of trade negotiations.
"We're asking people in areas where perhaps the party they normally vote for hasn't got a chance to win to lend their vote to another candidate who could beat the Conservatives."
Mr Packham added that they did not want to ignore people who wanted to leave the EU, so they would like a second referendum
"We didn't think that people were wrong when they voted leave," he added. "We think that people didn't know what Brexit actually meant, we do now."
Richard Sadler, chairman of North Yorkshire for Europe, said: "For goodness sake, don't be taken in by the lies and cons being put upon us by the Conservative Government. They are trying to make out that they will get Brexit done and everything will be fine and we can get on with our lives.
"The truth of the matter, as Boris Johnson well knows, is that if we were to go ahead with his disastrous Brexit deal, or no deal Brexit, there will be acrimony, arguments and division about Brexit for years and years to come.
"We are trying to get people to realise that now is the time to act for the long term future of our country and we must put the whole issue of Brexit, to resolve it, back to the people."
He added: "Please, please vote tactically to stop the scorched earth Brexit policies of the Prime Minister and vote for the candidate who's most likely to beat the Tory candidate.
"Do you really trust the Conservatives after all the years they've been in power to look after our public services, our workers' rights, our environmental protections, all the things we have fought for for years to build up?
"Yorkshire for Europe is not pushing any particular political party, but we are saying check the tactical voting websites and check the candidate who is most likely to beat the Tories to stop the disaster of the reckless Brexit policy that is being pursued."