Yes it is permissible to help the pet if the pet is in sever pain when there is no other to help it but euthanasia.
'Wassalam.
Sayyed Mohammad al-Musawi is originally from Iraq and heads up the World Ahlul Bayt Islamic League in London. Other than being involved in various humanitarian projects, he frequently responds to religious questions. In the past, he has also spent significant time in India guiding the community.