In modern day, a simple umbrella will protect you from rain, and a parasol will protect from the heat of the sun. The ability to protect yourself from the weather in a lot of countries and cultures has always been some what of a status symbol.
The Umbrella / parasol in Buddhism is also seen as a form of protection, but in a more symbolic spiritual way. The umbrella is typically depicted to be above the head of an important deity as a way to symbolise that beneath the umbrella is the centre of the universe. Some consider the umbrella to be like a mobile temple.
There are many different meanings and thoughts to the umbrella / parasol across different cultures and communities. In Tibet, people of importance and entitled to different types of parasol depending on their rank or importance, with the Dalai Lama being entitled to multiple types.
The Umbrella / parasol brings a sign of honour and respect, but above all symbolises the protection over the shadow that it casts.