It's not strictly correct to say that a planet orbits
its parent star.

Given that each exerts a gravitational pull on the other,
how could we expect the star to remain absolutely fixed
while the planet orbits around it?

In fact both orbit; and they orbit around a common point
called the Center of Mass (CM)

Wikipedia defines the Center of Mass (CM) as,

a specific point at which, for many purposes, the
system's mass behaves as if it were concentrated

Physicists have used the concept of "balance" to define CM,

The center of mass is familiar to anyone who has
ever played on a see-saw. The fulcrum point at
which the see-saw will exactly balance two people
sitting on either end is the center of mass for the
two persons sitting on the see-saw.

The figure below depicts a two-body system;
X marks its center of mass.


                                                         image credit

If X were the fulcrum of a see-saw, the two bodies
would balance.

For a two body system, the center of mass can be
determined through these equations: where m denotes
mass; d, distance; and R, the total separation between
the two bodies.

         
                                                         image credit


When the a planet and star orbit their center of mass,
CM will almost certainly be located within the star itself,
but not at its center, as shown in the video below.