Given a parametric lattice with a basis given by polynomials in $\Bbb{Z}[t]$, we give an algorithm to construct an LLL-reduced basis whose elements are eventually quasi-polynomial in $t$: that is, they are given by formulas that are piecewise polynomial in $t$ (for sufficiently large $t$), such that each piece is given by a congruence class modulo a period. As a consequence, we show that there are parametric solutions of the shortest vector problem and closest vector problem that are also eventually quasi-polynomial in $t$.