Jingle-bell is a compound noun, it can be plural or singular.
In the first phrase you have a noun "rock" being modified by an adjunct noun "jingle-bell". "Jingle-bell rock" isn't a sentence, nor is it a clause, as there is no verb. It is a headline meaning "This is 'jingle-bell' rock(-and-roll)"
The second line then uses "subject verb" sentences, with plural subject "jingle-bells" and a verb in simple present "ring".
We tend to use singular for noun adjuncts, for example, "chicken soup" not "chickens soup".
Here is a song by me
Chicken soup, chicken soup
Chickens go in the pot
For my chicken soup.
My chicken runs away
From me today
'Cos I want some chicken soup!
That isn't a great song... but notice that "chicken soup" is a singular noun-noun phrase, not a "subject verb" clause. But "Chickens go" and "my chicken runs" are both examples of "subject-verb". This is the same as "jingle-bell rock"