  In this paper we study the sections of the canonical line bundle on the
moduli space of parabolic semistable vector bundles with trivial determinant
and fixed parabolic structure of type $\underline{\lambda}=(\lambda_1,...,
\lambda_s)$ (with each weight $\lambda_i$ in $P_{\ell}(\SL(r))$) on a smooth
projective irreducible curve over $\C$ of genus $g \geq 1$. We give a
nontrivial lower bound for the dimension of the sections (that are called
generalized parabolic SL(r)-theta functions) when $\sum_{1}^{s} \lambda_i$ is
in the root lattice.
