April Long says they felt excluded when answering the 2021 survey with their partner Kelly given there was no question about sexual orientation.
The 65-question census also asked a difficult question about the rainbow family's baby Kaison.
"Kaison has two mums - I'm Mumma and Kelly is Mummy - but the form asked where Kaison's mother and father were born," Long said on Friday.
"Our initial reaction was shock. We were unable to complete it accurately. It didn't capture us. It made us feel invisible and it didn't count us."
Long, a chief executive for a non-profit addiction program provider, said they filed a complaint on Thursday with the Human Rights Commission alleging the failure to properly count LGBTIQ+ people in the census amounted to unlawful discrimination.
Equality Australia, which joined as a co-complainant, said Long's experience was replicated thousands of times with other LGBTIQ+ people and their families on census night.
"The fact is we still don't know how many LGBTIQ+ families there are in Australia, nor where they are located," Equality Australia legal director Ghassan Kassisieh said in a statement.
The complaint alleges the ABS and the assistant treasurer - then Liberal frontbencher Michael Sukkar - engaged in deliberate conduct that meant the statistics bureau "could not follow its own guidance" on the collection of data on sexual orientation, gender identity and sex characteristics.
The ABS and current assistant treasurer Stephen Jones have been contacted for comment.
Chief statistician David Gruen has previously said the bureau was instructed by the government to not ask about sexual orientation or gender.
"There will be an opportunity to revisit that for the 2026 census and the ABS will be engaging in a public consultation process starting later this year to ask the community if there are other questions people think we should be asking," he said in June.