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Let $Y$ count unordered pairs of distinct $K_4$ copies that have a common vertex. Classifying such a pair by whether it shares one, two, or three vertices givesIndeed, after substituting $p=n^{-2/3}\log n$, these terms are respectively

The first moment method gives $\mathbb{P}(Y>0)\le \mathbb{E}Y\to 0$. Part (i) says that $X\ge 100$ with probability tending to one, so the union bound shows that, with probability tending to one, both $X\ge 100$ and $Y=0$. On that event any one hundred of the $K_4$ copies are pairwise vertex-disjoint, proving vertex-disjoint sparse clique copies in this case.