Notation? Inverse of demand function
As a result we have $$ z\{p(t)\}\beta(t) = F\{K(t),L(t)\}. $$ Solving
  for the output price we obtain $$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$


I am pretty sure that what is meant here is that there is a $p(t)$ trajectory for which
$$
z\{p(t)\} = F\{K(t),L(t)\} / \beta(t)
$$
(see first equation)

Given the function $z$, this depends only on the right hand side of the equation, hence let us denote the trajectory $p(t)$ as a function of this, i.e.
$$ p(t) = p[F\{K(t), L(t)\}/\beta(t)]$$
A different notation would be
$$
p(t) = z^{-1}\left( F\{K(t), L(t)\}/\beta(t) \right).
$$
This second notation is useful because all the partial derivatives follow from this notation and the chain rule.