Did the Stoics invent propositional logic, and why was it forgotten for centuries?

Largely, yes: Chrysippus created a system of propositional logic, building on earlier work, and it was then forgotten for centuries. For around five hundred years Stoic logic was one of the two great systems of logic, discussed alongside Aristotle's. Aristotle's logic dealt with relations between terms like "Socrates" or "man". Stoic logic dealt with relations between propositions like "it is day", as in "if it is day, it is light; but it is day; so it is light".

The roots lie in the Megarian school, where Diodorus Cronus and his pupil Philo worked on conditional propositions; Zeno himself studied under the Megarians. Chrysippus seems to have introduced the three main connectives: "if", "and" and "or". Their "or" was exclusive, unlike the inclusive "or" generally used in modern formal logic. Among their basic argument forms are what we now call modus ponens (if p then q; p; therefore q) and modus tollens (if p then q; not-q; therefore not-p).

Why was it forgotten? Aristotle's logic prevailed, partly because it was seen as more practical and partly because the Neoplatonists took it up, and Chrysippus's logical writings are almost entirely lost. Knowledge of Stoic logic as a system stayed lost until the 20th century, when logicians familiar with the modern propositional calculus reappraised the ancient accounts. Before that, the 19th-century historian Carl Prantl called it "dullness, triviality, and scholastic quibbling" and welcomed that the works of Chrysippus no longer survived.

So the modern parallels were largely rediscovered, not inherited, though one scholar now argues a thread may have passed through Prantl to Frege.