All scripture has authority in the life of those who are in the Body of the Messiah, but some scripture has more authority than others. There is a hierarchy of authority between different sections of scripture that must be respected. Failure to respect this hierarchy will result in having to resort to mental gymnastics to explain anomalies (which are almost non-existent) as well as very wrong interpretations. What's also important to note is that the Greek word in the "New Testament" that is translated as "scripture" can refer to absolutely anything that is written down, including the writings which God inspired or dictated.
The foundation of Scripture is the Torah, which literally translated means "instruction." The Torah (often translated as "Law") comes directly from God and is made of the books of Genesis, Exodus, Leviticus, Numbers and Deuteronomy. It is the standard for testing anything claimed to be scripture, and anyone claiming to be a teacher or prophet. Anything that claims to be Scripture must be in agreement with what the Torah says. If any alleged Scripture contradicts the Torah or teaches against it, then that "scripture" is false scripture. Any prophet that contradicts or teaches against the Torah is a false prophet. And any teacher that contradicts or teaches against Torah is a false teacher.
After Torah, we have the Prophets, whom God spoke through. The Prophets have the second highest authority of any scripture because what they say is dictated directly by God. Prophets are always tested against the instructions contained in Torah. Prophets are also not allowed to contradict each other. Note that the Prophets spend a majority of their time calling people back to God's Torah. The importance of the Law and the Prophets over all other scripture is alluded to many times in the New Testament including Matthew 5:17, 7:12 and Acts 24:14 to name a few.
After the Prophets, we have the Writings, with the Psalms having the highest authority of the Writings (alluded to in Luke 24:44 as prophecy) and the Chronicles having the lowest authority of all the Writings. Like the Prophets, the writings are tested against Torah. But they must be tested against the Prophets as well. The majority of the writings are historical accounts, including accounts of previous prophecies being fulfilled. The reason that the Chronicles have the lowest priority of all the other writings is because it is a retelling of the historical accounts that were written before it.
That is the chain of authority of the books contained within what is falsely called the "Old Testament." The "New Testament" is not above the rules set forth above and must be tested against both the Torah and the Prophets. Like the books of the "Old Testament," the books of the "New Testament" have a chain of authority that must be followed. Since Jesus is God in the flesh, the accounts of what he did on the earth (referred to as "the Four Gospels") take the highest priority over the other "New Testament" books. The "New Testament" books that have the second highest priorities would be the ones written by those who were with Jesus during his time on the earth. Finally, Paul would have the lowest authority of all of the "New Testament" authors. Traditionally, Paul is held as if he has almost as much (if not more) authority as Jesus himself. But even he admits that Jesus appeared to him last "as one who is born abnormally" [1 Corinthians 15:8; NIV]. Paul was the only "New Testament" author who was not with Jesus during his earthly ministry.