1. Reading the Gospels as Protestants
This verse from the Aramaic scriptures tells us the virgins (we, Christians) will meet both the bridegroom and bride.
A few things here: firstly only a very small number of Protestants would accept Aramaic-primacy. It also doesn't have any relevance to this question. Just because the Greek text doesn't include "Again Jesus said to his disciples" doesn't mean that he was speaking to someone else. Moving on...
Second, when Jesus tells a parable we have to be careful to not draw too much out from it. Most parables have just one, maybe two, main points they are trying to teach. Protestants do not generally participate in allegorical interpretation where every detail corresponds to some spiritual reality.
Third, we have to note that Jesus was speaking to Jews prior to the cross. It is anachronistic to call his audience the Church or Christians. Much of what he says does of course apply to Christians, but not all. We have to carefully consider what he says in that original context before applying it to ourselves.
2. The Church as the Bride of Christ
Protestants believe this metaphor for the Church is taught in these passages:
Ephesians 5:25-27, 31-32 (NIV): Husbands, love your wives, just as Christ loved the church and gave himself up for her to make her holy, cleansing her by the washing with water through the word, and to present her to himself as a radiant church, without stain or wrinkle or any other blemish, but holy and blameless. ... “For this reason a man will leave his father and mother and be united to his wife, and the two will become one flesh.” This is a profound mystery—but I am talking about Christ and the church.
2 Corinthians 11:2-3 (NIV): I am jealous for you with a godly jealousy. I promised you to one husband, to Christ, so that I might present you as a pure virgin to him. But I am afraid that just as Eve was deceived by the serpent’s cunning, your minds may somehow be led astray from your sincere and pure devotion to Christ.
And in Revelation, John calls the Church the New Jerusalem, and then the New Jerusalem the Bride of Christ:
Revelation 3:12-23 (NIV): The one who is victorious I will make a pillar in the temple of my God. Never again will they leave it. I will write on them the name of my God and the name of the city of my God, the new Jerusalem, which is coming down out of heaven from my God; and I will also write on them my new name. Whoever has ears, let them hear what the Spirit says to the churches.
Revelation 21:2 (NIV): I saw the Holy City, the new Jerusalem, coming down out of heaven from God, prepared as a bride beautifully dressed for her husband.
Revelation 21:9-10 (NIV): One of the seven angels who had the seven bowls full of the seven last plagues came and said to me, “Come, I will show you the bride, the wife of the Lamb.” And he carried me away in the Spirit to a mountain great and high, and showed me the Holy City, Jerusalem, coming down out of heaven from God.
Note that in the Old Testament Israel is also frequently pictured as the wife of God, but Protestants differ over to whether the Church's relationship to Israel should be seen as a continuation, replacement, or parallel group of God's people.
3. Interpreting those parables
Matthew 9:14-15 (NIV): Then John’s disciples came and asked him, “How is it that we and the Pharisees fast often, but your disciples do not fast?”
Jesus answered, “How can the guests of the bridegroom mourn while he is with them? The time will come when the bridegroom will be taken from them; then they will fast.
In this metaphor Jesus says that, like Ecclesiastes says, "there is a time for everything". Fasting is one practice of spiritual devotion, but not the only one. Just as you don't fast during a wedding banquet, Jesus didn't think his disciples should fast while they were with him. But in the future, either while he was dead or after he ascended, it would be appropriate for his disciples to fast.
Should we make anything of the fact that his disciples are the guests rather than the bride? I don't think so, for as I wrote before, we are not allegorists who must find the spiritual parallel to every detail. But even if you do want to see a significance here, it doesn't negate the Bride of Christ metaphor. Because the New Testament uses many metaphors to describe the relationship between Christ and Church. A married couple is one, but another is that of the head and body of Christ. Are we to say that the Church is therefore married to itself because it is the body of Christ? Of course not. We don't mix metaphors; each one stands by itself to illustrate a different aspect of our relationship with God.
Matthew 25:1: At that time the kingdom of heaven will be like ten virgins who took their lamps and went out to meet the bridegroom.
Matthew 25:13: Therefore keep watch, because you do not know the day or the hour.
This parable tells us what its own meaning is: we have to watch and wait carefully for the "the sign of your coming and of the end of the age" (Matthew 24:3).
I think it's possible in this parable that the virgins/bridesmaids also refer to the Jewish people, some of which were waiting eagerly for their messiah, and some of which were not.
And lastly, some Dispensationalists see Israel as an enduring category even into the New Earth. Those who died as faithful Jews before Christ are the people of God, but not the Church, the Bride of Christ. If our lives after resurrection are seen as one giant wedding reception then those faithful Jews are the guests while the church is the bride. But this is a minority position within Protestantism.