The Parameter Estimates table is reporting p-values for a test where the null hypothesis is that one regression coefficient is zero. In the Type III tests, the null hypothesis is that a certain linear function of the regression coefficients is zero. Sometimes these two tests overlap, i.e. the linear function being tested is that just one coefficient is zero, in which case the Wald test statistic is calculated similarly.
The "Four Types of Estimable Functions" chapter in the SAS/STAT documentation provides some relevant information about the Type III tests. In particular, the very last line in the "Estimability" section covers the form of the Wald test statistic and the "Type III Estimable Functions" section covers how a hypothesis for the test is constructed. I will admit, whenever I have to refer to the material about constructing the tests I have to read it (at least) two or three times before it starts to sink in.
As to why the tests using the Wald statistics versus the generalized score statistic differ in your example, I'm not sure there's really much I can say. The number of clusters is not small but not so large to make differences between the tests surprising.
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