** Revised version giving an explicit integration of the equations:**

Consider the system of three PDE for two functions $f(s,t)$ and $g(s,t)$
$$
f_{ss}f_{tt}-{f_{st}}^2 = g_{ss}g_{tt}-{g_{st}}^2 = {f_t}^2+{g_t}^2-1 = 0.
$$
I am going to show that the general solution depends on two functions of one variable and give an explicit local parametrization of solutions on an open
domain in the $st$-plane.

Say that a (local) solution on a simply-connected open domain in the $st$-plane is *nondegenerate* if, when one writes $(f_t,g_t) = (\cos u, \sin u)$, one has $\mathrm{d}u\not=0$. In this case, because the other equations imply
$$
0 = (f_{ss}f_{tt}-{f_{st}}^2)\,\mathrm{d}s\wedge\mathrm{d}t = \mathrm{d}f_s\wedge\mathrm{d}f_t = -\sin u\, \mathrm{d}f_s\wedge\mathrm{d}u
$$
and
$$
0 = (g_{ss}g_{tt}-{g_{st}}^2)\,\mathrm{d}s\wedge\mathrm{d}t = \mathrm{d}g_s\wedge\mathrm{d}g_t = \cos u\, \mathrm{d}g_s\wedge\mathrm{d}u
$$
it follows that $f_s$ and $g_s$ must also be functions of $u$
(which is a function of $s$ and $t$ on the domain in question).
Let us assume that we have restricted to a subdomain $U$
in the $st$-plane on which the
fibers of the submersion $u:U\to\mathbb{R}$ are connected.
Let $I = u(U)\subset\mathbb{R}$ be the image interval.

I claim that the equations above imply
$$
\cos u\,\mathrm{d}f_s + \sin u\,\mathrm{d}g_s = 0.
$$
This follows because, if $\mathrm{d}u = p\,\mathrm{d}s + q\,\mathrm{d}t$,
then $p$ and $q$ do not simultaneously vanish (since $\mathrm{d}u\not=0$),
and the equations $\mathrm{d}f_t = f_{st}\,\mathrm{d}s + f_{tt}\,\mathrm{d}t$
and $\mathrm{d}g_t = g_{st}\,\mathrm{d}s + g_{tt}\,\mathrm{d}t$
then imply
$$
f_{tt} = -q\,\sin u,\quad f_{st} = -q\,\sin u,\quad
g_{tt} = p\,\cos u,\quad g_{st} = p\,\cos u.
$$
Then the equations $f_{ss}f_{tt}-{f_{st}}^2 =g_{ss}g_{tt}-{g_{st}}^2 =0$,
coupled with the fact that $\sin u$ and $\cos u$ only vanish on isolated
level curves of $u$, imply that
$$
q\,f_{ss} = -p^2\,\sin u\qquad \text{and}\qquad q\,g_{ss} = p^2\,\cos u.
$$
Since $p$ and $q$ do not simultaneously vanish, it follows that $q$
cannot vanish and, hence, that
$$
f_{ss} = -(p^2/q)\,\sin u\qquad \text{and}\qquad g_{ss} = (p^2/q)\,\cos u.
$$
This establishes the claim. In particular, it follows (since the
fibers of $u$ are connected) that there is a function $h:I\to\mathbb{R}$ such that
$$
\mathrm{d} f_s = -h(u)\,\sin u\,\mathrm{d}u
\qquad \text{and}\qquad
\mathrm{d} g_s = h(u)\,\cos u\,\mathrm{d}u.
$$

Now, because
$$
\begin{align}
\mathrm{d} f
&= f_s\,\mathrm{d}s + f_t\,\mathrm{d}t= \mathrm{d}\bigl(sf_s + t f_t\bigr)
- \bigl(s\,\mathrm{d}f_s + t\,\mathrm{d}f_t\bigr)\\
&= \mathrm{d}\bigl(sf_s + t f_t\bigr) + (s\,h(u) + t)\,\sin u\,\mathrm{d}u,
\end{align}
$$
and, similarly,
$$
\begin{align}
\mathrm{d}g
&= g_s\,\mathrm{d}s + g_t\,\mathrm{d}t= \mathrm{d}\bigl(sg_s + t g_t\bigr)
- \bigl(s\,\mathrm{d}g_s + t\,\mathrm{d}g_t\bigr)\\
&= \mathrm{d}\bigl(sg_s + t g_t\bigr) - (s\,h(u) + t)\,\cos u\,\mathrm{d}u,
\end{align}
$$
one sees, by applying $\mathrm{d}^2=0$, that one must have
$$
\mathrm{d}(s\,h(u){+} t)\wedge \sin u\,\mathrm{d}u
= \mathrm{d}(s\,h(u){+} t)\wedge \cos u\,\mathrm{d}u = 0.
$$
Thus, $(s\,h(u){+} t)$ must be a function of $u$ alone, i.e.,
$s\,h(u){+} t = k(u)$ for some function $k:I\to\mathbb{R}$.

Now, set $v = t\,h(u) - s$, so that one can solve for $s$ and $t$ in
terms of $u$ and $v$ as
$$
s = \frac{k(u)\,h(u)+v}{\bigl(1+h(u)^2\bigr)}
\qquad\text{and}\qquad
t = \frac{k(u)-v\,h(u)}{\bigl(1+h(u)^2\bigr)}.
$$
It then follows, because $\mathrm{d}s\wedge\mathrm{d}t\not=0$,
that $(u,v):U\to\mathbb{R}^2$ is a coordinate system on $U$.

To complete the integration, set $h(u) = a(u) + a''(u)$.
This determines $a:I\to\mathbb{R}$ uniquely up to the addition
of a constant linear combination of $\cos u$ and $\sin u$,
and, thus, one immediately sees that there is a unique choice of $a$ so that
$$
f_s = a(u)\,\cos u - a'(u)\,\sin u
\qquad\text{and}\qquad
g_s = a(u)\,\sin u + a'(u)\,\cos u.
$$
Similarly, choosing a function $b:I\to\mathbb{R}$ so that $k(u) = b(u)+b''(u)$,
one has
$$
\mathrm{d}\bigl(b(u)\,\cos u - b'(u)\,\sin u)
= -\bigl(b(u)+b''(u)\bigr)\,\sin u\,\mathrm{d}u
= -k(u)\,\sin u\,\mathrm{d}u
$$
and
$$
\mathrm{d}\bigl(b(u)\,\sin u + b'(u)\,\cos u)
= \phantom{-}\bigl(b(u)+b''(u)\bigr)\,\cos u\,\mathrm{d}u
= \phantom{-}k(u)\,\cos u\,\mathrm{d}u.
$$
Thus, there is a unique choice of $b$ so that
$$
f = s\,f_s + t\,f_t - \bigl(b(u)\,\cos u - b'(u)\,\sin u\bigr)
$$
and
$$
g = s\,g_s + t\,g_t - \bigl(b(u)\,\sin u + b'(u)\,\cos u\bigr).
$$
With the formulae for $f_s$ and $g_s$ above and the expanded formulae
$$
s = \frac{\bigl(b(u){+}b''(u)\bigr)\,\bigl(a(u){+}a''(u)\bigr)+v}
{\bigl(1+\bigl(a(u){+}a''(u)\bigr)^2\bigr)}
\qquad\text{and}\qquad
t = \frac{\bigl(b(u){+}b''(u)\bigr)-v\,\bigl(a(u){+}a''(u)\bigr)}
{\bigl(1+\bigl(a(u){+}a''(u)\bigr)^2\bigr)},
$$
one now has formulae for $s(u,v)$, $t(u,v)$, $f(u,v)$, and $g(u,v)$
in terms of two arbitrary functions $a$ and $b$ of $u$ and their first
two derivatives.

Conversely,
given a choice of $C^2$ functions $a$ and $b$
on an interval $I\subset\mathbb{R}$, on the open set
in $I\times\mathbb{R}$ on which $\mathrm{d}(s(u,v))\wedge\mathrm{d}(t(u,v))
\not=0$, the surface
$$
\bigl(s(u,v), t(u,v), f(u,v), g(u,v)\bigr)
$$
in $\mathbb{R}^4$ is (locally) a graph of the form
$$
\bigl(s, t, f(s,t), g(s,t)\bigr)
$$
where $f$ and $g$ satisfy the desired $3$ equations.

Moreover, every nondegenerate solution is locally of the above form.

Note that $s(u,v)$, $t(u,v)$, $f(u,v)$, and $g(u,v)$ are all
linear in $v$, which shows that the above surface in $\mathbb{R}^4$
is *ruled*, i.e., is a $1$-parameter family of lines, which, apparently,
is what the OP meant by asking whether there was a 'concatenated parametrization'.

Finally, let us say that a solution $(f,g)$ is *totally degenerate* if $\mathrm{d}u\equiv0$. In this case, of course, $f_t$ and $g_t$ are constants and the above analysis does not apply. However, the general totally degenerate solution is
$$
(f,g) = \bigl(a(s) + (\cos u)\,t, b(s) + (\sin u)\,t\bigr),
$$
where $a$ and $b$ are arbitrary functions of one variable and $u$ is a constant. Thus, this represents a different class of solutions that depend on two functions of one variable.

Outside of a closed set with no interior, the graph of a general solution is a union of graphs of nondegenerate solutions and totally degenerate solutions.