Hotel La Victoria saw no problem with La Orotava parking for us. They told us to leave the rental car outside their premises. This was a solution in case we didn't find a space in the old quarters near Casa de los Balcones. This important Tenerife landmark and attraction is just around the corner from the antique hotel in the old quarters of the town.
A route from the TF5 in direction Mayorazco and then past the Orotava police head quarters up the hill towards Iglesia de Nuestra Señora de Concepción will also get you to find our parking.
Otherwise, you might spot a space for your vehicles in the old quarters by simply driving around its blocks of old mansions and houses. In previous years, we used to be lucky above the Casa de los Balcones by the ancient town hospital.
Looking by the bridge near the gardens Jardines de Victoria and Plaza de la Constitución is pretty useless.
More chances are at the modern Mayorazco township. However, you need to get back up a steep hill to reach the old quarters again, unless you want to see the museums near the Mayorazco first.
The narrow, steep roads of the old town during Corpus Christie celebrations prove why this section isn't for La Orotava parking. An alternative chance which applies to any parking on Tenerife is to type a famous tourist attraction in the vicinity of your destination into your smart phone to get parking help.