That's not a continuity error, that's an aspect of Spock's character.

• In "Amok Time" Kirk learned that Spock was engaged to T'Pring, and he also comments that Spock never mentioned how important his family was after learning that T'Pau would be officiating the ceremony.

• In "Journey to Babel" Kirk suggests to Spock that he might want to travel to the surface of Vulcan to spend time with his parents, while Sarek and Amanda are standing right in front of them. Kirk learns that Spock's father is one of the most well regarded diplomats in the Federation.

• In "Yesteryear" when Spock returns from this journey to the Vulcan of his childhood, he tells Kirk that the only thing that changed was that a pet died. He doesn't mention that it was his childhood pet that died protecting him.

• In "Star Trek: The Final Frontier" Kirk learns that Spock had a half brother, Sybok. At that point they've known one another for 22 years.

• In "Sarek" Picard mentions having met Sarek at Sarek's son's wedding. What son? Spock? If it was Spock, there's no mention of him having a wife when he later shows up in the "Unification" two parter, or in the Kelvin films. Another, as of yet, unnamed member of Spock's family?

It would be a bigger continuity issue if Spock had mentioned a sister.

I noped out in season 1 because I didn't think Discovery was woke enough