Hint: The difference between $P$ and the mirror reflection $x^1 \mapsto -x^1$ in the $(x^2,x^3)$-plane is a $\pi$-rotation around the $x^1$-axis, cf. comment by Meng Cheng.
Recall that the Lorentz group $O(3,1)$ has 4 connected components $$ O(3,1)~=~SO^+(3,1) ~\cup~ P~SO^+(3,1)~\cup~ T~SO^+(3,1) ~\cup~ PT~SO^+(3,1). $$ The connected components containing the identity is the restricted Lorentz group $SO^+(3,1)$.