Cut a piece of parchment paper to fit inside a 4- to 5-qt. pot. Find a small heatproof plate that also fits inside pot, and set paper and plate aside.
In pot, combine 6 cups water, the sugar, and quinces. Cover pot and bring almost to a simmer. Carefully set parchment, then plate on top of fruit to weight it down. Cook at a gentle simmer without disturbing until quinces are tender and pale pink in color, 1 1/4 to 1 1/2 hours (lift plate with a table knife to check). Discard parchment and let quinces cool in syrup.
Make ahead: Up to 3 weeks, chilled.