Does fifth postulate of Euclid imply the existence of parallel lines?

Yes.

According to Euclid's 5th postulate, when n line falls on l and m and if

, producing line l and m further will meet in the side of ∠1 and ∠2 which is less than

If

The lines l and m neither meet at the side of ∠1 and ∠2 nor at the side of ∠3 and ∠4. This means that the lines l and m will never intersect each other. Therefore, it can be said that the lines are parallel.