No, this isn't what the Bible says, and I find nothing in the Bible that forbids cremation as a means of disposing of a person's body.
It's true that burial was the common practice in the Bible, and cremation was rare. When cremation was practiced, it showed contempt for the person (e.g., Achan, who disobeyed God and brought defeat to the Israelites—see Joshua 7:25). Cremation is often practiced today in cultures that have no respect for the human body or see it as evil; Christians in those societies reject cremation. We believe God gave us our bodies, and they should be treated with respect. After creating the human race, "God saw all that he had made, and it was very good" (Genesis 1:31).
One reason burial has been preferred by Christians is because the Bible teaches that one day those who die in Christ will be raised from the dead and given new bodies. But God is able to bring together whatever has been scattered: "And he will send his angels and gather his elect from the four winds, from the ends of the earth to the ends of the heavens" (Mark 13:27).
I hope your family won't be divided over this issue. Pray for peace in your family—and may each of you face the reality of death, and rejoice in the hope we have of eternal life because of Christ's death and resurrection for us.