In a similar question,
Equation of angle bisector, given the equations of two lines in 2D,
someone was given a task similar to yours, except that in their case it was explicitly specified that if the lines were parallel the line halfway between them should be given as the answer.
In an answer to the same question,
a formula was given for the angle bisectors of any two intersecting lines
with equations
\begin{align}
a_1x + b_1y + c_1 &= 0, \\
a_2x + b_2y + c_2 &= 0.
\end{align}
To write the answer in a more compact format, let
\begin{align}
q_1 = \sqrt{a_1^2 + b_1^2}, \\
q_2 = \sqrt{a_2^2 + b_2^2}.
\end{align}
Then the equations of the two angle bisectors are
\begin{align}
(a_1 q_2 + a_2 q_1)x + (b_1 q_2 + b_2 q_1)y + c_1 q_2 + c_2 q_1 &=0, \tag1\\
(a_1 q_2 - a_2 q_1)x + (b_1 q_2 - b_2 q_1)y + c_1 q_2 - c_2 q_1 &=0. \tag2
\end{align}
There are two such lines because the original two lines form two pairs
of vertical angles and each of the bisectors bisects just one pair of angles.
The two angle bisectors are perpendicular to each other.
In the case of two parallel lines,
\begin{align}
ax + by + c_1 &=0, \\
ax + by + c_2 &=0,
\end{align}
Equation $2$ has zero coefficients for both
$x$ and $y$ (and therefore no longer describes a line),
while Equation $1$ becomes
$$
2ax + 2by + c_1 + c_2 = 0, \tag3
$$
which is the equation for a line midway between the two given lines.
I find this to be an interesting "limit" property, but not a justification
for the answer given for your practice problem.
In my opinion it is misleading to call the answer for parallel lines an "angle bisector," and the problem should have been posed in the manner of
Equation of angle bisector, given the equations of two lines in 2D
instead. But it may be customary on the exam you're preparing for that
the "midline if the lines are parallel" clause is implicitly understood
to be part of any angle bisector question.
I would regard this as a quirk of the exam--a very bad quirk in my opinion, adding a completely unnecessary reason why one would need coaching for such exams, but that's a complaint for another forum, perhaps.
(You probably have little choice at this time other than to accept the existence of such quirks and learn to deal with them.)
I would not regard this quirk as an application of analytic geometry as most practitioners understand it.
