"Foundations of Projective Geometry" by Hartshorne says the following:
The completion of the affine plane of four points is a projective plane with 7 points.
The affine plane of $4$ points is essentially a paralellogram $ABCD$. The completion will contain $A,B,C,D,[AB],[AD],[AC],[BD]$. Here $[AC]$ is the point of intersection of all lines parallel to $AC$ with the line at infinity (in other words it is an ideal point).
Hence I am getting $8$ points instead of $7$. Where am I going wrong?
To paraphrase what has been said so that the question is answered:
There are only three distinct pencils of parallel lines: $\{\overleftrightarrow{AB}, \overleftrightarrow{CD}\}$, $\{\overleftrightarrow{AC}, \overleftrightarrow{BD}\}$ and $\{\overleftrightarrow{AD}, \overleftrightarrow{BC}\}$. These give rise to three ideal points in the projective completion.
The mistake made was thinking $\overleftrightarrow{AC}$ was not parallel to $\overleftrightarrow{BD}$ apparently because of a visual illusion of intersecting diagonals while modeling the geometry as the corners of square.
We can also count everything in another way. Since each line has two points, the underlying field is $F_2$. The projective plane then corresponds to one dimensional subspaces of $\mathbb F_2^3$, of which there are $8-1$ (there are seven distinct generators of $1$ dimensional subspaces, since $0$ will not work.)