It seems that this problem is studied under the heading of Lamé families, i.e., those families of surfaces belonging to an orthogonal triple. Cayley was the first to give a general characterization of Lamé families in terms of a system of 3rd-order differential equations -- a simplified form can be found on p. 466 of A. R. Forsyth, "Lectures on the Differential Geometry of Curves and Surfaces.". In his paper, "On Lamé Families of Surfaces" , (1926), C.E. Weatherburn gives an even simpler characterization via the following condition on the unit normal field $n$:
$$ \mathrm{div}(\frac{\partial}{\partial n} \mathrm{curl} mathrm{curl}\ n) = 0.$$
The
In the paper, the author applies this condition as though if it is were sufficient to characterize a Lamé family (see, e.g., the paragraph on p. 303 following the statement of the theorem), but from the proof it seems like only a necessary condition. In any case, in the book, "Differential Geometry of Three Dimensions" the same author states,
"A necessary and sufficient condition that a family of surfaces may form part of a triply orthogonal system is that, on each surface, the vector $\bar{\nabla}\psi$ be the gradient of some scalar function." (p. 101)
Here $\psi = ||n||$, and the operator $\bar{\nabla}$ is given by $\bar{\nabla}\phi = -\nabla \phi \cdot \nabla n$ where $\nabla$ is the surface gradient operator. In particular (he notes), any parallel family of surfaces is a Lamé family. So the answer to the original question is positive (since the distance function $f_1$ defines a parallel family).
