As j.c. mentioned in his answer, the average distance between parallel supporting planes is better known as the *mean width*. More generally, one can take a convex body in $\mathbb{R}^n$ and consider the average $k$-volume of its projections to $k$-dimensional linear subspaces (the case $k=1$ is the mean width). These quantities are called *quermassintegrals*, and they are proportional to the coefficients in the Steiner formula for the volume of the parallel body. Your question is about the case $n=3$, $k=1$ of this theorem.

To prove it, start with $n=3$ and $k=2$. This is Cauchy's theorem which says that the area of the boundary of the convex body is one quarter of its average projection area:
$$
Area(\partial K) = \frac1{\pi} \int\limits_{\xi} Area(pr_\xi K)\, d\xi,
$$
where the integral is taken over all unit vectors $\xi$, and $pr_\xi K$ is the projection of $K$ to $\xi^\perp$. The proof is very nice, just integrate the projection area of each face individually.

Now replace $K$ in the Cauchy formula by the $K_\epsilon$, the $\epsilon$-neighborhood of $K$ and apply the Steiner formula to both sides. The LHS is easy to compute by decomposing the boundary of $K_\epsilon$ into flats, pieces of cylinders, and pieces of spheres:
$$
Area(\partial K_\epsilon) = Area(\partial K) + \epsilon\sum_e \ell_e(\pi-\delta_e) + \epsilon^2 \cdot 4\pi.
$$
For the RHS observe that $pr_\xi(K_\epsilon) = (pr_\xi K)_\epsilon$ and expand the integral in a similar way:
$$
Area(pr_\xi K_\epsilon) = Area(pr_\xi K) + \epsilon L(\partial pr_\xi K) + \epsilon^2 \cdot \pi.
$$
Now integrating and comparing the coefficients at $\epsilon$ we obtain
$$
\sum_e \ell_e(\pi-\delta_e) = \frac1\pi \int_\xi L(\partial pr_\xi K)\, d\xi
=\frac1\pi \int_\xi \frac12 \int_{\eta \in \xi^\perp} L(pr_{\xi\oplus\eta}K)\, d\eta d\xi,
$$
where we applied the Crofton formula relating the length of a curve to the average length of its projections. It remains to transform the right hand side:
$$
\frac1\pi \int_\xi \frac12 \int_{\eta \in \xi^\perp} L(pr_{\xi\oplus\eta}K)\, d\eta d\xi = \int_\zeta L(pr_{\zeta^\perp}K)\, d\zeta,
$$
and we get the integral of lengths of projections of $K$ to lines.

This proof comes from 1920's, I guess, and the procedure is called Kubota's recursion. For exact references see Notes for Section 4.5 of

*Schneider, Rolf*, Convex bodies: the Brunn-Minkowski theory, Encyclopedia of Mathematics and Its Applications. 44. Cambridge: Cambridge University Press. xiii, 490 p. (1993). ZBL0798.52001.