  Let $\tilde{\mathfrak{g}}$ be the affine Lie algebra of type $A_{2l}^{(2)}$.
The integrable highest weight $\tilde{\mathfrak{g}}$-module $L(k\Lambda_0)$
called the standard $\tilde{\mathfrak{g}}$-module is realized by a tensor
product of the twisted module $V_L^T$ for the lattice vertex operator algebra
$V_L$. By using such vertex algebraic construction, we construct bases of the
standard module, its principal subspace and the parafermionic space. As a
consequence, we obtain their character formulas and settle the conjecture for
vacuum modules stated in arXiv:math/0102113.
