To show parking areas, click the Layers button and select Parking in the menu that opens (to hide them, just click it again).
The Parking button only works in areas where Yandex.Maps has information about parking.
Types of parking are denoted by color:
Blue — free public parking
Blue with the national currency symbol — public paid parking (dark blue ─ expensive, light blue ─ less expensive)
Red — areas where parking is forbidden: