In the book of Euclidean and Non-Euclidean Geometry by Greenberg, it is given that
DEFINITION: Lines l and m are parallel if they are distinct lines and no point is incident with both of them.
DEFINITION: A projective plane is a model of incidence geometry having the elliptic parallel property (any two lines meet) and such that every line has at least three distinct points lying on it (strengthened In cidence Axiom 2).
DEFINITION: An affine plane is a model of incidence geometry having the following Euclidean parallel property:
For all line $l$, and for all points $P$ that are not incident to $l$, there exists a unique line $m$ s.t $P$ is incident to $m$ and $m$ and $l$ are parallel.
[...] So the idea in extending an affine plane to a projective plane is to add enough new "points at infinity" so that all lines parallel to any given line will now meet at one such point.
However, by definition two parallel line cannot have a common point which is incident to both, and when we add "new points at infinity" to make all line meet at a single point, we will not have any parallel lines anymore, so we cannot be working in affine geometry anymore.
Moreover, isn't the definition of affine plane same as the parallel postulate of Euclid ?
Edit: In case I misunderstood someting, here is the exact formulation: ($\exists !$ means exits uniquely)