Probably inspired by trees, the first umbrellas could be huge leaves for example. In general, with history development, we see that it is the king's privileged article, and no one but the king could have one.
As a symbol of worldly power, the umbrella has taken on the proportions, colorations and embellishments that were inherited from ancient religious traditions.
Hindu Shveta-chattra was a symbol of supreme law and the symbol of supreme authority placed over his head at his coronation. So, it also was transferred on the highest terrestrial power as the instrument of protective and sheltering firmament. Here, society and religion are intractably related. And the earthly sovereign was seen as the exemplar and custodian of Dharma, universal responsibility. The Ramayana depicts the ceremonial in which the sage Vasishtha installed Rama upon a golden throne while Shatrugna placed a white umbrella over his regal head. Also, that white umbrella emerged from churning of the ocean and was given to Varuna. In rituals, especially dedicated to Vishnu, the temple umbrella is used very often.
In Christianity, the Pope used a ceremonial parasol umbrella as ceremonial regalia. Also known as the pavilion, in modern usage the umbraculum is a symbol of the Roman Catholic Church and the authority of the Pope over it. It is found in the contemporary Church at all the basilicas throughout the world, placed prominently at the right of their main altars.
In Buddhism, it symbolises the center of the universe and spiritual support. Umbrellas can also represent a kind of mobile temple. And Buddha will often display a large and elaborate umbrella above his head. When Buddha blessed his disciples with the beatific vision of Buddha-fields, they saw stretching before them myriads of Buddhas serenely seated under trees and umbrellas.
In Vajrayana Buddhism, this large umbrella (atapatra) was even deified into the thousand-armed-footed goddess Sitapatra, whose name literally means the white umbrella.
In Jainism, umbrella (chhatra ) is one of auspicious symbols also. You can see the divine three-tiered parasol above Tirthankara's head. As one of the insignias of royalty, and in a metaphorical sense, it represents the spiritual power. The Jains hold that the entire manifested universe may be represented as a colossal human form topped by an umbrella symbolizing the ultimate heaven, conveying that every man is capable of spanning the realm of the mediators.
Wat Phra That Doi Suthep, Thailand
Well, it's possible to search and to find analogies still long.From ancient megaliths...
to Madur Madanantheswara Siddhi Vinayaka Temple, for example.
The general idea that umbrella is symbol of universal Dharma and the vertical handle represents the central axis that upholds the world. If the umbrella is over an important person, it indicate that the person or symbol below is the center of the universe. The space an umbrella is the space between heaven and earth. And then, it can be a symbol of transfer of divine law between God and man. Personally, to unlock the mystery of the umbrella, we should seize its handle for to pass its energy upwards imitating the MahaYogin Shiva, who is the Master. And then, that energy can graciously go down to top of a head and to heart. After all, all ribs of umbrella ultimately converge in the center.
OM NAMAH SHIVAYA