In their answer @Soporiferous correctly says of the relation between the semipartial correlation and the standardized beta coefficient, and hastes to label my old comment to the question (about the sooner existence of a relation between the beta and partial correlation) "slightly misleading". But in my comment, I implied Beta of another regression (another dependent variable) than @Soporiferous seemingly implied.
Let us have 3 variables X, Y, Z.
Partial correlation between X and Y (Z partialled out from both) is
$r_{xy.z} = \frac{r_{xy} - r_{xz}r_{yz} }{\sqrt{ (1-r_{xz}^2)(1-r_{yz}^2) }}$.
While semipartial or part correlation between X and Y (with Z partialled out from Y) is
$r_{x(y.z)} = \frac{r_{xy} - r_{xz}r_{yz} }{\sqrt{1-r_{yz}^2 }}$.
@Soporiferous correctly notices (by linking to an outer source) that the last formula is very similar to the formula of a beta regression coefficient:
$\beta = \frac{r_{xy} - r_{xz}r_{yz} }{{1-r_{yz}^2 }}$, with the only difference being is that root taken in the denominator.
True observation; however mind that this beta is $\beta_y$ in the regression where X is the dependent while Y and Z are predictors. Semipartial $r_{x(y.z)}$ squared is equal to the rise of R-square in reaction to the inclusion of Y to the model having consisted only of Z.
So, the semipartial is structurally similar to $\beta_y$, - and not to $\beta_x$ (of regression where Y is the dependent which idea would probably come to mind at first place "by default").
But partial $r_{xy.z}$ is related to both the $\beta_y$ (where X is the dependent) and to the $\beta_x$ (where Y is the dependent):
$\beta_x = \frac{r_{xy} - r_{xz}r_{yz} }{{1-r_{xz}^2 }}$.
Saying (in my comment to the question and in my answer linked just now) that regression coefficient beta is directly related to partial correlation
I meant that regression where X is a regressor and the partial correlation considered is, too, between the regressor X and the dependent Y. The very title of the question induces such interpretation of it.