In 2 Peter 3:9b, we see God’s heart on the subject: “He is patient with you, not wanting anyone to perish, but everyone to come to repentance.” But, no, there is no specific promise found in Scripture that guarantees salvation for a believer’s family. Each person must choose, individually, to serve Jesus Christ. Each one must realize his sinfulness, go before the Lord in humble repentance, and ask Jesus to be Lord of his life.
The good news is that the Lord put you into their lives! He has burdened you with the desire to see them become saved; He has equipped you with the Word of God so that you can be a witness for Christ; and He has sent the Holy Spirit ahead of you to begin drawing your loved ones unto Himself. In the meantime, you need to pray constantly for them, live as Christ before them and let God love them through you. Ask the Lord to give you opportunities to present the Gospel–in His timing and with the leading of the Holy Spirit. As Jesus said, “Let your light so shine before men, that they may see your good works and glorify your Father in heaven” (Matt. 5:16).