Andy Miller and Brian Stephens fell in love 12 years ago while they were training for a marathon on the trails around Lady Bird Lake in downtown Austin. As they began talking about starting a family, they knew of only one same-sex couple who had successfully adopted. “There weren’t a lot of role models,” Mr. Miller said. Mr. Miller and Mr. Stephens adopted their son, Clark, days after he was born in 2007. But on Clark’s birth certificate, only Mr. Miller’s name appears under “father.” “Mother” remains noticeably blank — and Mr. Stephens’s name is not listed. Texas law prevents the names of two gay parents from being on supplemental birth certificates for their adoptive children. The forms provide space for only one mother, a woman, and one father, a male. The gender-specific language was added to the Texas Health and Safety Code in 1997 as part of a renewed commitment to conservative values, said its author, former State Representative Will Hartnett, Republican of Dallas. Opponents of the provision say it compels same-sex families to present unwieldy paperwork to prove legal parentage for medical care, school enrollment and international travel, and prompts extra scrutiny that can embarrass or confuse children.