My dog had an illness and has been able to recover enough to walk. However, it's different from before since unfortunately the condition still has him (her) paralyzed from the left side.
ご質問ありがとうございます。
「左半身に麻痺が残る」はそのまま英語で「still paralyzed from the left side」と言います。
また、コンテキスト的に「愛犬が病気になり今は歩けるまで回復したけど、前とは違く、左半身に麻痺が残った状態なの」と言いたいなら、そうすると、「My dog had an illness and has been able to recover enough to walk. However, it's different from before since unfortunately the condition still has him (her) paralyzed from the left side.」という風に説明できます。
ご参考になれば幸いです。
アメリカ合衆国 
