Rome does not have the burden of proving that the sources outside of Scripture are equally authoritative, because of the simple fact that the other parts of the Word of God draw their authority from the same source as the Bible, namely, the Catholic Church. The Bible has any weight at all only because the Catholic Church, by Her authority, declares that it does. This same authority of the Catholic Church also declares Sacred Tradition to be of weight.
It is not the burden of Rome to prove that Sacred Tradition is of equal weight to the Bible, but rather it is the burden for those outside of Rome to prove that the Bible has any authority itself. The Catholic Church claims to be a Divine institution of God endowed with the authority to teach by God Himself. By this authority, She accepts the cannonical books of the Bible as authoritative. By this same authority, She also accepts Sacred Tradition as so.
However, those whom do not profess to be of the Catholic Church have no evidence to support their claim that the Bible is authoritative at all. Does the Bible itself say so? Surely not, and, if it did, this fact would be of no value, for one cannot prove the authority of something from within itself. The Qu’ran, too, claims to be authoritative, and yet Protestants do not regard it so, for this is not proof. Joseph Smith claimed to be a prophet of God, and yet this is not proof that he truly was. So too would a claim within the Bible to be authoritative fall short of the burden of proof.
Protestants often declare the Bible’s authority to rest in the fact that it contains no error. However, this is not proof of its authority either. For inerrancy proves only only itself, not authority. Further, in appealing to the inerrancy of the Scriptures, one must first prove that they are in fact inerrant, a task which can be done only to the extent as those things which are contained therin are testable, and, while much of that so contained is testable, much is not. Indeed, should one wish to appeal to Biblical inerrancy for proof of its inerrancy, one must prove the existence of God, the immortality of the soul, indeed, even the existence of the soul. These tasks are those which cannot be accomplished in this life.
In short, the burden lies not on Rome to prove Sacred Tradition. Rather, the burden rests on all of those whom reject Rome to prove the authority of the Bible at all.