'das Paar' is a noun meaning 'couple' or 'pair' - strictly two objects. 'paar' is an quantifier meaning 'couple' or 'few' - can be two or more, and quantifies a noun, not necessarily relating to humans.
So use the noun version when refering to a 'couple' as in a man and wife, or a pair, such as a pair of shoes ('ein Paar Schuhe')
Use the quantifier version where you would say 'a few' or 'a couple'. e.g a couple of problems, a few oranges.
The earlier explanation by hutcho66 made this very clear for me. If the intent had been "a few people are swimming", the German phrase would have to be "Ein paar Menschen schwimmen" (or Ein paar Mädchen or Ein paar Freunden or Ein paar Jungen, etc.) When used by itself and with a capital "P", "Paar" is a stand-alone noun, considered singular. Since "das Paar" is singular, you can't use schwimmen, which can only be used with plural nouns. Thanks to hutcho66, I now understand "das Paar" to mean two people with a connection. It can also mean two things that belong together (like gloves or shoes). This is corroborated in the Duden which defines "das Paar" as ...
Substantiv, Neutrum - 1a. zwei zusammengehörende oder eng miteinander …1b. zwei [als Männchen und Weibchen] …2. zwei zusammengehörende Dinge
Note: zusammengehörende = together + belonging
In English, 'couple' is a collective noun, so you can treat it as either singular or plural, meaning you can say 'The couple are swimming' or 'The couple is swimming', both are fine. The former emphasizes the two people as individuals, while the latter emphasizes the couple as a unit.
In German, because 'Das Paar' is a strictly singular noun, so you must treat is as an 'it' (as it is neutral).
The translation seems to be wrong, 'ein Paar' = A couple, 'ein paar' = a couple of, a few. 'Das Paar' is a noun meaning 'the couple' or 'the pair', 'paar' is a quantifier which always goes with 'ein' to get 'a few' or 'a couple of''. EXACTLY the same way the word 'couple' works in English.
It's probably worth reporting if you notice things like this.
You can tell from context here. all we have is "Ein Paar schwimmt", so "Ein Paar" has to be the subject of the verb. The only meaning of "Ein Paar" that makes sense there is for it to be article + noun = "A couple". It couldn't make sense as article + adjective here.
(If you see it written it is more obvious, as it is is "Paar" (noun) not "paar" (adjective), but you can tell just from the spoken version)
Is "Ein Paar" as a noun, with a capital P, and with a 3rd-person-singular verb, strictly "a couple" as in "a romantic couple"? Or could this sentence be the answer to a question like, "What are the ducks doing?" Because in English, if you asked, say, about a couple on their honeymoon in Hawaii, you could say "A couple is swimming", whereas the answer to the duck question would be "A couple ARE swimming." So yeah...