Is a parent more important than a testamentary guardian? Who has priority over the other?
Generally, when you have a kid by default you are the natural guardian of the kid.
A Testamentary Guardian is one who is appointed by a will of the natural guardians of the minor.
Technically speaking, there's nothing written that says that one has priority over the other. So if say you appoint your brother as a guardian for your son to act jointly with your wife, both your brother and your wife are guardians to your son. Both parties have the same rights with regards to performing their duties to the child as the guardians.
A common question is if the testamentary guardian can act on behalf of the infant without the consent of the natural guardian? Ideally, it's always better to seek the consent of all guardians (including testamentary guardians) with regards to important decisions.
In terms of signing off documents (e.g. buying insurance policies for minors), you may want to check with the individual institution(s) if they accept the document with one party's signature without the other.
In the event there's disagreement between the two guardians, it may end up in Court with the results on a case-by-case basis due to the many factors involved, with the child's welfare one of the most important factors.
While it should not be generalized indefinitely, a recent case published by Straits Times "Woman's appeal over change in daughter's surname and race dismissed by High Court", Judicial Commissioner Tan Puay Boon "rejected the mother's argument that parental authority takes precedence over a guardian's authority in such disputes".
Understandably, parents might want to appoint a guardian only if there are no surviving parents (e.g. both parents die in a car crash.)
You can do so by selecting the appropriate clause under the "Appoint People" page.