Is a wife obligated to remain with her husband who practices homosexuality, under the pretext of parental consent and seeking reward from God, or is she obligated to separate from him and live her own independent life?

Sodomy is one of the grave sins and immoral acts, and its perpetrator deserves punishment. If your husband is doing this, do not despair of advising him, warning him of Allah's punishment, helping him strengthen his connection with his Lord, and associating with the righteous. However, if he insists on committing this grave sin, you are not obligated to obey your family in staying with him. It is better for you to seek a divorce from him, for "Allah will enrich each out of His abundance."