It also says the PiS manipulates TV coverage.
The European Commission (EC) has asked the Polish government for changes.
Demonstrators, many from other parts of the country, waved Polish and European Union (EU) flags and sang the national anthem as they gathered in Warsaw's Bank Square.
"We are for a democratic Poland, for a European Poland, for a Poland that is proud, that seeks friends and partners, not enemies in Europe as it is today," PO leader Grzegorz Schetyna, told protesters waving Polish and European Union flags.
However PiS leader Jaroslaw Kaczynski denied freedom was at risk in Poland.
"Today we do have freedom and by taking part in this march and claiming that today freedom is at stake, as a matter of fact, you are marching in the name of the opposite," he said during a visit to the city of Szczecin.
The EC is due to present its report on the situation in Poland later this month.
It wants the Polish government to reverse curbs on the constitutional tribunal, which it said endangered the rule of law.
It has also said new government powers to appoint the heads of public TV and radio may jeopardise EU values.
However, BBC correspondents say there is little support in the bloc for punitive measures against Poland.
PiS officials, meanwhile, have denied Civic Platform accusations that they want to take Poland out of the EU.
In March divisions between the Polish government and the EU deepened over the EU's re-election of former Polish prime minister and Civic Platform leader Donald Tusk as EC president.
Mr Kaczynski holds Mr Tusk "politically" responsible for the 2010 plane crash in Russia which killed his twin Lech Kaczynski, then Poland's president, and all other 95 people on board.
Many Poles believe Mr Tusk's government did not do enough to explain the causes of the crash, which took place in dense fog.
Poland is due to hold municipal elections next year. Recent polls showing the PO ahead of the PiS for the first time since 2015.
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