Do you pay to enter the national park on the Czech side of the Karkonosze?

No. As Karkonosze National Park (KPN) points out, no entry fees are charged on the Czech side of the mountains (Krkonoše National Park, KRNAP). So you can see, for example, Mumlava Waterfall in Harrachov without a ticket — only car parks may be paid. On the Polish side, within KPN, the entry ticket applies (11 PLN, reduced 5.50 PLN).