The riddle
What Goes Up When the Rain Comes Down?
Two directions at once
Rain falls, and the thing that answers it is opened and lifted overhead, so the question sets two movements against each other. The riddle states the contrast and leaves the listener to find something raised to meet the weather. Tellers give the answer as umbrella or umbrellas, and both count. A canopy of fabric on ribs, held up on a pole, is how Wikipedia describes the object.
A word that began with shade
Wikipedia traces umbrella to the Latin umbra, meaning shadow or shade, and notes that the Oxford English Dictionary records the first use in that sense in 1611. The oldest umbrellas were meant to keep off the sun. The French parapluie joins para, to shield, with pluie, rain, and parasol does the same with the word for sun. English settled on one word for the cover in either weather.
When the cloth was waxed
Wikipedia reports that the rain umbrella began to appear in France in the 1660s, when the fabric of sun umbrellas was coated with wax. A lightweight folding model that opened and closed like a modern one was introduced in Paris in 1710 by a merchant named Jean Marius, and it weighed less than one kilogram.
An icon for rain
Wikipedia notes that in weather forecasts an umbrella is the standard icon for rain, and that it has a symbol of its own in Unicode, which means it can be typed as a character. A different riddle about an ordinary household thing is what gets wetter the more it dries.