Common distributions usually fix their skewness. Beta distribution has two parameters to determine all of the mean, variance, and skewness.
Student-T's skewness can change by some definitions but it does not have a dedicated parameter.
What distribution has exactly three parameters for mean, variance, and skewness?
Infinitely many answers are possible, some links in comments. One family is the skew-normal, but it only admits limited degrees of skewness. The same ideas used to construct the skew-normal family from the normal family can be used for other families!
For empirical modeling with skewness have a look at the gamlss system, which admits many families.
