These slotted brass rings are unquestionably from parasols and umbrellas.
They have been around for a very long time, and in use far and wide across the world.
Personally I have seen and dug hundreds of these on sites all over the country, from mixed Colonial/early 19th Century, right on up to late Victorian. They have been found on remote island beaches at sea level, and mountain top cabin sites at over 10,000 feet! I've dug them in military sites, home sites, tent sites, and places one might think nobody had ever set foot. They are found in downtown construction sites along busy streets, and lonely places in the middle of deserts, where rain rarely falls. Basically, anywhere men, women, and children, lived or traveled in the 19th Century, can produce these brass parasol and umbrella rings.
I'm not precisely sure how early these commonly found brass rings date. Umbrellas and parasols have though, been around for a very long time, even hundreds of years or more. Without delving into more precise research on time-frame, and merely going by excavation experience, I will wager the majority of these slotted brass rings came into use following the Industrial Revolution, when machine manufacturing allowed for larger scale production of goods at reduced costs.
http://www.bigsiteofamazingfacts.com/history-of-umbrellas
CC Hunter