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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?

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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.

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