In my field both the parallel test and Velicer's minimum average partial (MAP) test are commonly used when researchers are considering how many factors to retain in factor analysis or components to retain in PCA.
I find the parallel test reasonably intuitive - we want factors/components that account for more variance than factors/components derived from random data.
As I understand it, in the MAP test we do a PCA/factor analysis with only one factor/component, and then partial that PC/factor out of the correlations between the observed variables, and then we calculate the average squared partial correlation coefficient from the off-diagonals of the partial correlation matrix. Then we repeat the process with two factors/components, and so on. Then we look for the number of factors/components that results in the lowest average squared correlation coefficient.
On some level I think I understand the mechanics of the process described above, but I don't understand how it relates to selecting an appropriate number of factors/principal components. The original paper is short and fairly readable, but I did not feel after reading it that I understood the concept intuitively.
Velicer, W. F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41(3), 321-327.