Longleat Safari Park, near Warminster, Wiltshire, said it had no choice but to put down lioness Louisa and her cubs because they displayed "odd aggressive behaviour".
A sixth lion was destroyed on "welfare grounds" after being injured when he was attacked by his brother and Louisa last month.
News of the decision to put down the lions caused fury on Longleat's Facebook page. Gordon Smith wrote: "This seems on the surface a shameful decision."
Longleat said Louisa's behaviour was due to "poorly managed genetic history prior to her arrival" in 2011 aged 18 months and that she and the cubs showed signs of neurological problems.
The park said it would have been irresponsible to move the animals elsewhere.