We consider two closely related problems of text indexing in a sub-linear working space. The first problem is the Sparse Suffix Tree (SST) construction of a set of suffixes $B$ using only $O(|B|)$ words of space. The second problem is the Longest Common Extension (LCE) problem, where for some parameter $1\leτ\le n$, the goal is to construct a data structure that uses $O(\frac {n}τ)$ words of space and can compute the longest common prefix length of any pair of suffixes. We show how to use ideas based on the Locally Consistent Parsing technique, that was introduced by Sahinalp and Vishkin [STOC '94], in some non-trivial ways in order to improve the known results for the above problems. We introduce new Las-Vegas and deterministic algorithms for both problems. We introduce the first Las-Vegas SST construction algorithm that takes $O(n)$ time. This is an improvement over the last result of Gawrychowski and Kociumaka [SODA '17] who obtained $O(n)$ time for Monte-Carlo algorithm, and $O(n\sqrt{\log |B|})$ time for Las-Vegas algorithm. In addition, we introduce a randomized Las-Vegas construction for an LCE data structure that can be constructed in linear time and answers queries in $O(τ)$ time. For the deterministic algorithms, we introduce an SST construction algorithm that takes $O(n\log \frac{n}{|B|})$ time (for $|B|=Ω(\log n)$). This is the first almost linear time, $O(n\cdot poly\log{n})$, deterministic SST construction algorithm, where all previous algorithms take at least $Ω\left(\min\{n|B|,\frac{n^2}{|B|}\}\right)$ time. For the LCE problem, we introduce a data structure that answers LCE queries in $O(τ\sqrt{\log^*n})$ time, with $O(n\logτ)$ construction time (for $τ=O(\frac{n}{\log n})$). This data structure improves both query time and construction time upon the results of Tanimura et al. [CPM '16].