  In this paper we define the parametric Korteweg-de Vries hierarchy that
depends on an infinite set of graded parameters $a = (a_4,a_6,\dots)$. We show
that, for any genus $g$, the Klein hyperelliptic function
$\wp_{1,1}(t,\lambda)$ defined on the basis of the multidimensional sigma
function $\sigma(t, \lambda)$, where $t = (t_1, t_3,\dots, t_{2g-1})$, $\lambda
= (\lambda_4, \lambda_6,\dots, \lambda_{4 g + 2})$, determines a solution of
this hierarchy, where the parameters $a$ are given as polynomials in the
parameters $\lambda$ of the sigma function.
  The proof uses results on the family of operators introduced by V. M.
Buchstaber and S. Yu. Shorina. This family consists of $g$ third-order
differential operators of $g$ variables. Such families are defined for all $g
\geqslant 1$, the operators in each of them commute in pairs and also commute
with the Schr\"odinger operator.
  In this paper, we describe the relationship between these families and the
parametric Korteweg--de Vries hierarchy. A similar infinite family of
third-order operators on an infinite set of variables is constructed. The
results obtained are extended to the case of such a family.
