Linton was the one who put down the two animals.
Burton said she was disappointed because she had offered to re-home the mother and calf to save them from being killed.
The animals were allowed to roam near the park's recreational trails and farm manager Linton said there had only ever been one other confrontation in the past 18 years.
Auckland Council head of parks services Mark Bowater said staff visited Totara Park first thing on Monday morning and provided assistance to Linton.
"We worked with them to identify an individual cow being overly protective of her calf. During calving season, cows are typically very protective of their calves and do not like people to get near them," Bowater said.
"We recommend anyone who chooses to go in parts of our parks with cows is extra careful during calving season, as it is a cow's natural instinct to protect their calves. Over this time people should keep their distance from cows, and ensure they never come between them and their calves."