The parasol, in other words, an umbrella is a traditional buddhism symbol of royalty and protection from the raging heat of the tropical sun. The coolness of its shade signifies shield from the aching heat of suffering, temptation, hindrances, illnesses, and harmful forces. An umbrella can protect people from the different elements, like the sun or the rain. In this context, a parasol or umbrella can mean protection from suffering and harmful forces. It can also mean the enjoyment of the cool shade it provides.
Just like the Bodhi tree is the sanctuary from which the Buddha attain enlightenment, so is the parasol through which people enjoy the cooling shade of protection from Buddhas and bodhisattvas.