SALT LAKE CITY — The congregants wore church clothes to the annual gay pride parade here in the heartland of Mormonism. Waving rainbow signs and urging love and acceptance, they marched at the front of the line, vanguards of a new movement of Mormons challenging their church’s staunch opposition to homosexuality.

In 2008, the Church of Jesus Christ of Latter-day Saints played a pivotal role in financing and supporting a ballot measure that outlawed same-sex marriage in California. Now, as the legal battle over that ban, Proposition 8, heads toward the Supreme Court, gay and straight Mormons are making increasingly vocal calls for church leaders to reconsider their stance on gay marriage and welcome openly gay congregants back into the church.

“I hope the church will have a change in doctrine, that the membership will have a change of heart,” said Diane Stewart, 58, a Mormon who teaches Sunday school in Salt Lake City. “If they can’t wrap their arms around it, they really will be left behind.”

This gentle and scattered dissent, percolating in Utah living rooms, online message boards and gay pride parades across the country, could mark the beginnings of a small earthquake in the socially conservative Mormon Church, coming at a moment when Mitt Romney’s presidential bid and President Obama’s support for gay marriage have focused national attention on both the Mormon Church and the politics of same-sex marriage.