Here are the relevant passages from the NASB:
The disciples went and did just as Jesus had instructed them, and brought the donkey and the colt, and laid their coats on them; and He sat on the coats.
They brought the colt to Jesus and put their coats on it; and He sat on it.
They brought it to Jesus, and they threw their coats on the colt and put Jesus on it.
Jesus, finding a young donkey, sat on it;
It appears as though Christ only sat upon the colt, not both the colt and the donkey. Mark, Luke, and John didn't even bring up the donkey in their reiteration of the events of that day, apparently finding it's presence not worth mentioning. If Christ had ridden on both of them at once, it would have been quite an odd site, and it would seem to be something that all four gospel writers would consider important, especially as all four of them do mention the colt and the general event in question.
Both Matthew (verse 5) and John (verse 15) give us some background of the reason for Christ entering Jerusalem on the back of a donkey's colt. It is the fulfillment of the prophecy in Zechariah 9:9.
Rejoice greatly, O daughter of Zion! Shout in triumph, O daughter of Jerusalem! Behold, your king is coming to you; He is just and endowed with salvation, Humble, and mounted on a donkey, Even on a colt, the foal of a donkey.
Thus, the primary point of importance for the passage in Matthew seems to be to demonstrate and validate the claim that Christ is indeed the prophesied King of Israel. Thus, regarding 2 Timothy 3:16, this scripture is profitable for TEACHING that Christ is King, for REPROOF of the claim that He never said He was, for CORRECTION of those who were unaware that He is the fulfillment of prophecy, and for TRAINING IN RIGHTEOUSNESS in that it identifies the promised Messiah -- He who is the Righteous One (Isaiah 53:11), the One we should be imitators of (1 Corinthians 11:1; Ephesians 5:1).