I think the thread is almost on target except my thought is it is just slightly off since it is ..The Parasol ..not an especially Hindu sign but deeply rooted in Buddhism as one of the
8 auspicious signs..see
https://www.ancient-symbols.com/buddhist-symbols.html the Parasol not only gives protection from the elements of rain and sun but protects on a spiritual belief level (thus Talisman) from evil and disease. therefor like a Talisman does ;it steps across religious boundaries through tradition and tribal acceptance as do several western talisman signs and beliefs and so sits very comfortably with Hindu myths and legends.
On the Tulvar Hilt it occurred to me that the POMMEL bears resemblance to the parasol and this is posed as a general question...Is it? See also the sketch above with the man sitting with a Tulvar under a Parasol..for comparison.