If you want an example which is an "officially named parameterized distribution family, you can look into the generalized gamma distribution, https://en.wikipedia.org/wiki/Generalized_gamma_distribution. This distribution family has three parameters, so you can fix mean and variance and still have freedom to vary higher moments. From the wiki page, the algebra do not look inviting, I would rather to do it numerically. For statistical applications, search this site for gamlss, which is an extension of gam (generalized additive models, in itself a generalization of glm's) which have parameters for "location, scale and shape".

Another example is the $t$-distributions, extended to be a location-scale family. Then the third parameter will be te degrees of freedom, which will wary the shape for a fixed location and scale.

If you want an example which is an "officially named parameterized distribution family, you can look into the generalized gamma distribution, https://en.wikipedia.org/wiki/Generalized_gamma_distribution. This distribution family has three parameters, so you can fix mean and variance and still have freedom to vary higher moments. From the wiki page, the algebra do not look inviting, I would rather to do it numerically. For statistical applications, search this site for gamlss, which is an extension of gam (generalized additive models, in itself a generalization of glm's) which have parameters for "location, scale and shape".

Another example is the $t$-distributions, extended to be a location-scale family. Then the third parameter will be the degrees of freedom, which will wary the shape for a fixed location and scale.