I am using DESeq2 to compare control samples to samples in which gene A has been either knocked down or overexpressed. Using the Wald test to compare control vs. knockdown and control vs. overexpression, I have found genes significantly impacted by both knockdown and overexpression of gene A.

However, I have seen a second method of using the likelihood ratio test, followed by a clustering tool (degPatterns) to identify genes following a specific expression pattern (i.e., knockdown < control < overexpression). The number of significant genes I receive through this method is much higher than for the Wald test. I'm quite new to RNA-seq analysis and am wondering if one method is preferred or more reliable.

Check the DESeq2 vignette section on LRT.

If you have multiple groups, the LRT is a global test for differences (similar to an F test for standard linear regression) whereas the Wald tests in DESeq2 are often pairwise comparisons (similar to t test).