We give algorithms that, given a straight-line program (SLP) with $g$ rules that generates (only) a text $T [1..n]$, builds within $O(g)$ space the Lempel-Ziv (LZ) parse of $T$ (of $z$ phrases) in time $O(n\log^2 n)$ or in time $O(gz\log^2(n/z))$. We also show how to build a locally consistent grammar (LCG) of optimal size $g_{lc} = O(δ\log\frac{n}δ)$ from the SLP within $O(g+g_{lc})$ space and in $O(n\log g)$ time, where $δ$ is the substring complexity measure of $T$. Finally, we show how to build the LZ parse of $T$ from such a LCG within $O(g_{lc})$ space and in time $O(z\log^2 n \log^2(n/z))$. All our results hold with high probability.