How do sola fide adherents explain The Parable of the Ten Virgins?

The short answer, is that works are not a requirement of salvation, but are an indication of salvation through faith, and the Holy Spirit working within the heart of the saved person. As such, the virgins in the parable that were not prepared were not of faith, as indicated by their surprise at the arrival of the bridegroom. Keep in mind that this parable is in reference to the second coming of Christ.

This interpretation is largely supported by the idea that the virgins were "foolish", as opposed to forgetful in failing to bring enough oil to wait until night, which demonstrates a lack of faith on their part that the bridegroom would arrive at all.

Also important is verse 25:11 of Matthew (ESV): "Afterward the other virgins came also, saying, 'Lord, lord, open to us.' But he answered, 'Truly, I say to you, I do not know you.'"

Note that he does not say that they were not prepared or had not done enough to merit entry, but that he did not know them at all, a phrase used by Jesus to indicate non-believers, such as the Pharisees in John 8:19 (ESV): "They said to him therefore, "Where is your Father?" Jesus answered, "You know neither me nor my Father. If you knew me, you would know my Father also." The Pharisees knew Jesus' identity, of course. He was speaking to knowledge of him as the Son of God.

The comments above are my interpretations. The following additional response is taken largely from the interpretations presented in the HCSB Study Bible, specifically Charles Quarrels of Louisiana College, who authored the commentary on Matthew.

In the HCSB commentary, the virgins represent those who fail to persevere by waiting for Jesus' return with constant vigilance. The cry of "Master, master" (Gk kurie, kurie ) is identical to that of the false disciples in Matthew 7:21 . "I do not know you" echoes Matthew 7:23 and expresses exclusion from Messiah's kingdom.

The parable does not describe a true disciple who loses his salvation, but a false one whose commitment to Jesus was deficient from the start.

People misunderstand this and other warnings in the NT because they see only two groups of people in the world: unsaved and saved. But according to 1 Corinthians 2:15 and 3:1-4, there are three groups: unsaved ("natural,") carnal, and spiritual. So, many of the warnings in Galatians and Hebrews are addressed to the carnal, for one aspect of the carnal mind is to trust in oneself and one's works rather than Christ.

So people see the warnings like the 10 Virgins and believe that the ones who missed out are unbelievers. Yet unbelievers do not wait for the Bridegroom! Rather, these were the carnal ones in the ecclesia (which are all saved by biblical definition), who did not heed the signs and get ready for His return. (See Matthew 24:42-51 for examples of readiness and unreadiness.) The spiritual are always ready, since Jesus is their focal point.

We don't really need to spiritualize the oil as being the Holy Spirit indwelling the believer; it is sufficient that the 5 are noted for being "foolish," not "without the Holy Spirit." They were foolish because they were not doing what should be done to get ready for his coming. If they had faith that he was coming, then they would have been ready with the oil in the first place. They believed enough to be saved and be put into the house of God, but not enough to stir themselves to action to readiness afterward. For possible examples as to why they were foolish, or unfruitful, see the Parable of the Sower (Matthew 13:18-33).

Faith alone saves us. The foolish missed the Bridegroom; but because they believed, they did not lose their salvation. They did not build on the foundation of Christ, and so their works will be seen to be wood, hay, and stubble, yet they "will be saved, yet so as through fire" (1 Corinthians 3:11-15). All the warnings to members of the ecclesia harmonize when you see this duality of flesh vs. spirit, carnal vs. spiritual, works vs. faith everywhere in the NT. The dire warnings are not about losing their salvation and they never say so.