I understand the Park test for heteroskedasticity has three different forms. The best known one is in a log form: LN(Residual^2) = intercept + slope (LN(X)). The second one is in a linear form: Residual^2 = intercept + slope (X). In both cases if the regression coefficient of X (the independent variable you are testing for heteroskedasticity) then you have to reject the null hypothesis that residuals are homoskedastic relative to the levels of the tested independent variable X. Nevertheless, do you know how well established is the second form, the linear one? And, also what is the third form? For model variables I need to test, it is key that I can use the linear form because many of those variables are percent changes that can't be logged.
I am less familiar with the Park test. The Wikipedia page only lists what you call the first form. What you are calling the second form is identical to the Breusch-Pagan test. It is very well established, for what that's worth. Regarding the distinction between using logged or non-logged predictors and response variables, it may help you to read this excellent CV thread: Interpretation of log transformed predictor. In general, the two-stage approach to modeling (i.e., test for assumptions, and fit standard model if non-significant or robust model if significant) is not recommended (see this excellent CV thread: A principled method for choosing between t test or non-parametric e.g. Wilcoxon in small samples). If you are worried about the possibility of heteroscedasticity, you would be better off just using robust methods, such as the Huber-White heteroscedasticity consistent 'sandwich' standard errors, by default. For some examples of various strategies that can be used with heteroscedastistic data, it may help to read my answer here: Alternatives to one-way ANOVA for heteroscedastic data.