Parking in La Orotava is a nightmare. If you’re not staying here it’s far better to just come by bus from Puerto de la Cruz, which is just 9km away. Bus 345 (€1.45, 20 minutes) leaves roughly every half-hour from 5.55am to 10.10pm, running to the bus station in the east of town.
If you have to drive, there's street parking here, but you should plan on arriving early to secure a spot.
Walking is also possible, as La Orotava is only around an hour from Puerto de la Cruz.