Thanks for sending along the links. It appears as if the R implementation computes the PACF by iteratively solving the Yule-Walker equations, which is perfectly valid. That said, it's not the only way to do it. As I mentioned, PARCORR computes the PACF by fitting successive order AR models by OLS, retaining the last coefficient of each regression, which is also perfectly valid, yet different.
Moreover, if I paste your 32-point dataset into the linked website, I can certainly see that R and MATLAB results differ, but I do think that the results are just that, "different", and I suspect neither is "wrong" give the method of calculation.
Also, recall that the Yule-Walker equations are based on the assumption of a stationary/invertible model, and your data clearly does not satisfy that assumption.
As I mentioned previously, I suspect that for data well-described by a stationary/invertible model the results should be very close asymptotically. To test this, I tried a 1000-pont random series, and got virtually identical results in MATLAB and R. And for small samples sizes I would expect the results to differ bit as well, even for well-behaved data.
Therefore, again this does not seem to be a bug, and the MATLAB results are not wrong, yet they are different.
As an additional reference please see Box, Jenkins, and Reinsel (1994), bottom of page 67.
Although they generally illustrate the calculation of PACF by solving the Yule-Walker equations they also say that "The partial autocorrelations may be estimated by fitting successive autoregressive models of orders 1, 2, 3, ... by least squares and picking out the estimates phi11, phi22, phi33, ... of the last coefficient fitted at each stage".
This is exactly what the MATLAB function PARCORR does.
Moreover, please notice that this Box & Jenkins section is in the chapter entitled "Linear Stationary Models."
I'll look into this a bit more and discuss it with the Econometrics Team to see if we can enhance PARCORR to use the Yule-Walker equations as an alternative, and if we find a bug we'll fix it. That said, this certainly does not seem to be a bug at the moment.