Here is one reason, the core of which can be found at page 17 of Moser's book mentioned in the comments. Many theorems about the heat equation are valid on "cylindrical" domains of the form $(0,T) \times \Omega$. If, in particular, $\Omega \subseteq \mathbb R^n$ is an open Euclidean ball $B(x_0,r)$ and $T = 2r^2$ (the same $r$), then

$$
\begin{align}
(0,T) \times B(x_0, r) =& \{ (t, x) \in \mathbb R \times \mathbb R^n ; | t - r^2 | < r^2 \text{ and } \|x - x_0\| < r \} = \\
=& \{ (t, x) \in \mathbb R \times \mathbb R^n ; \sqrt {| t - r^2 |} < r \text{ and } \|x - x_0\| < r \} = \\
=& \{ (t, x) \in \mathbb R \times \mathbb R^n ; \max \Big (\sqrt {| t - r^2 |}, \|x - x_0\| \Big) < r \} = \\
=& \{ (t, x) \in \mathbb R \times \mathbb R^n ; d \big( (t,x), (r^2, x_0) \big) < r \} = B_d \big( (r^2, x_0), r) \big)
\end{align}
$$

where $B_d (p, R)$ is the ball of centerpoint $p$ and radius $R$ with respect to the distance $d$.

The use of this metric turns cylinders into balls, and balls are always nice to work with.

Another motivation (bottom of page 18) is that many parabolic problems find a natural framework in the setting of Hölder spaces. You might know that on a metric space with distance $d$, $f$ is $\alpha$-Hölder continuous (with $\alpha \in (0,1]$) if and only if

$$|f(x) - f(y)| \le C d(x,y)^\alpha \ \forall x, y$$

for some $C>0$. It turns out that when studying the heat equation, the correct spaces are a bit different; namely, they are called "parabolic Hölder spaces" and are made of those $u$ for which there exists $C>0$ such that

$$|u(s,x) - u(t,y)| \le C \Big( \sqrt {|s-t|} + \| x-y \| \Big)^\alpha \ ,$$

which does not look like the Hölder condition give above. In reality, since

$$\sqrt {|s-t|} + \| x-y \| \le 2 \max \Big( \sqrt {|t-s|} + \| x-y \| \Big) = 2 d \big( (s,x), (t,y) \big)$$

the "parabolic Hölder condition" becomes

$$|u(s,x) - u(t,y)| \le C' d\big( (s,x), (t,y) \big)$$

with $C' = 2^\alpha C$, which looks exactly like the "usual" Hölder condition on metric spaces - which allows you to use the full theory of Hölder spaces.

Of course, the deep reason of this all is the fact that the space variables enter parabolic equations with derivatives of order $2$, while time only contributes a derivative of oder $1$. In order to bring them on equal footing and restore the broken symmetry, it is useful to resort to this metric trick.