Chatra, The umbrella (Skt. chattra; Tib. gdugs) is a traditional Indian symbol of protection and royalty.
The ability to protect oneself from the elements has always been, in all cultures, a symbol of status.
Umbrella in Buddhism = protection
In Europe, until just a few decades ago, an umbrella was a symbol of status for ladies of society. In Eastern thought, its protection of the bearer from the scorching heat of the sun was transferred into the religious sphere as "protection against the heat of impurities". Thus, the coolness of its shade symbolizes protection against the heat of suffering, desire, and other spiritually harmful forces.
The dome of the umbrella is held aloft by a vertical handle (much like how a mountain supports the sky), which is identified with the axis mundi, or central axis that supports the world.
The umbrella is held over an important dignitary or the image of a deity, to indicate that the person or symbol under the umbrella is indeed the center of the universe, and also its spiritual support. Umbrellas appear to be particularly significant in processional rites, serving as mobile temples. Thus, representations of the Buddha often show an elaborate and large umbrella above his head.
The chatra
As it is held over the head, it naturally symbolizes honor and respect. In Vajrayana Buddhism, this large umbrella (atapatra) has even been deified as the goddess Sitapatra with a thousand arms and legs, whose name literally means "the white umbrella".
In Tibet, depending on their status, various dignitaries were entitled to different umbrellas, with religious leaders entitled to a silk umbrella and secular rulers to a parasol with embroidered peacock feathers. Distinguished personalities like the Dalai Lama are entitled to both, and in processions, first a peacock parasol and then a silk parasol are carried after him.
The Tibetan version of the umbrella was adopted from its Indian and Chinese royal prototypes, and fashioned from a striped wooden frame with a bulging silk cover and hanging silk pendants forming an overhanging skirt.
The dome symbolizes wisdom, and the hanging skirt, compassion. The composite shape of the umbrella thus signifies the union of these two elements.
Octagonal and square umbrellas are also common, representing respectively the Noble Eightfold Path and the four cardinal directions.
Other Sources: https://en.wikipedia.org/wiki/Chatra
To learn more about Buddhist symbols, check out the complete article: BUDDHIST SYMBOLS AND THEIR MEANING
Video illustration of Buddhist parasol in full ceremony from the YouTube channel: Kevin Ly
