A set $\mathcal{P}$ of parabolas through a point $P$ not at infinity (it depends on notation used, but when $x_0\neq 0$), whose axis $r \notin P$, is a bundle of parabolas? If yes, what are its base points?
I am asking this because I am not sure about an excercise I did. Thinking to the semi-self-polar triangle (two points at infinity and the vertex as proper point) I would say not.
A parabola is the set of points which have equal distance from the directrix $d$ and the focus $F$. Here $F$ lies on the symmetry axis $r$ and $d$ is orthogonal to it. So if you know $P$ is a point on the parabola, then you can pick a line $d$ orthogonal to $r$ as the directrix. This gives you the distance $\lvert d,P\rvert$. Drawing a circle of this radius around $P$ intersects $r$ in up to two points. Either of them could be the focus $F$. So in this description you have one real degree of freedom, namely the choice of $d$, and one boolean degree of freedom picking either focus. And some constraints to ensure the circle intersects the line at all. Which you could avoid by picking $F$ first and $d$ second. Essentially the same situation.
But the boolean parameter can be avoided. You can think of the general operation of defining a conic through $5$ points. One of them is $P$, another $P'$ its mirror image in $r$. The point at infinity in the direction of $r$ is a contact point, so it has algebraic multiplicity $2$. Pick a fifth point anywhere in the plane and you get a conic from a one-parameter family of conics which is either the one you want or a superset thereof.
Pencils defined by four points can be described as linear combinations of degenerate conics through these. In the case at hand, a limit process can be used to describe the situation where two of the points are the same, turning the line joining them into the line at infinity as a tangent. To describe the family of conics explicitly let's concentrate on a special case. The setup is invariant under similarity transformations. So without loss of generality you can choose $r$ as the line $y=0$, $P=(0,1)$ and $P'=(0,-1)$. Or in homogeneous coordinates $r=[0:1:0]$, $P=[0:1:1]$, $P'=[0:-1:1]$. Then the family of conics can be described as
$$\lambda(y+z)(y-z)+\mu\,x\,z=0$$
For $\lambda=0$ this is the degenerate conic consisting of the line at infinity and the line joining $P$ to $P'$. For $\mu=0$ you get the lines parallel to $r$ through $P$ and $P'$. Both these degenerate conics correspond to situations where focus and directrix are at infinity. All other linear combinations lead to non-degnerate conics, and you will find a focus and directrix to describe them.
So the topology of the family is that of a punctured real line. Is that a bundle? Well, don't know your definition for that.