As you suggested, a water-party is a party on water - as can be seen from this 1812 chapbook cover.
Source: Victoria and Albert Museum. Note: Chapbooks "are small cheap publications of a popular nature purchased either at booksellers in towns or from chapmen (from early English 'ceap' meaning trade) and pedlars in rural areas."
This page of Regency Definitions describes a water-party as a social event
on a boat with pauses to view gardens
Perhaps the most famous water-party was when George Frideric Handel was commissioned by George I to compose Water Music
to accompany a vast water party organized by the court
for the evening of July 17 and 18, 1717
According to the newspaper the Daily Courant, the River Thames was practically covered with boats for this event, making it quite possibly the biggest ever water-party.
Painting of George Frideric Handel (left, with right arm extended) with King George I of Great Britain, traveling by barge on the Thames River while musicians play in the background. The painting is an artist's rendering of the first performance of Handel's Water Music in 1717. Attrib: By Edouard Jean Conrad Hamman (1819-1888) (P.M. History. Januar 2006, S. 29.) [Public domain], via Wikimedia Commons
Water-parties were one of the many possible 'activities of the ton' (pronounced 'tone'). The ton are described here as
High society; the elite; generally the wealthiest and those of rank,
with royalty at the top; in today's terms, the 'in' crowd, such as
Hollywood stars. To be "good ton" was paramount, and opened most any
door in fashionable society.
Other social events included the card-party, balls, the garden party (breakfast outdoors), the Venetian breakfast (an afternoon party) and a rout ("a crowded party with no music or dancing or places to sit but people went because it was a place to see and be seen").
The Georgian era in England is commonly used for the period 1714 to 1830/37 (so Georges I, II, III, IV + sometimes William IV 1830-7) and includes the Regency Era (strictly speaking, 1811–1820). The Scarlet Pimpernel is set in 1792, well within the Georgian era, so the reference is not anachronistic.
A water party is also mentioned in Jane Austen's Emma (published 1815/16):
"And then, he saved her life. Did you ever hear of that?--A water
party; and by some accident she was falling overboard. He caught her."
"He did. I was there--one of the party."
Another reference to water parties can be found in Sports in the Western World which mentions that members of the Cumberland Fleet (a yacht club founded in 1775 and named after George III's brother, the Duke of Cumberland)
...sponsored "water parties" on the Thames
The term 'water party' is still in use today, though with a far wider range of meanings and social groups (see, for example, Urban Dictionary and babble.com).