Why do Teddy's parents allow Rikki-Tikki-Tavi to stay in the house?

Teddy's parents allow Rikki-Tikki-Tavi to stay in the house because they recognize his bravery and the protective role he plays in safeguarding their family from dangerous snakes. After Rikki-Tikki saves Teddy from a cobra, they appreciate his value as a guardian and a companion. Additionally, they are aware that Rikki-Tikki, being a mongoose, has natural instincts that help keep their home safe from threats.