Writing an uncertainty of the form $8.3\pm1.2$ using 'parenthesis' notation similar to $5.9722(6)$

Both forms are correct, but the uncertainty is rarely if ever reported in an abbreviated form in cases like that of your example. The first form is definitely ambiguous and can be easily mistaken. The second one is less ambiguous, but not well known (it's correct anyhow, see the Guide to the expression of uncertainty in measurement, §7.2.2, point 3).

Thus, you're both right, but, generally speaking, I'd avoid ambiguous forms.