別れる
 

Hi, I would like to know if 別れる is only used when referring to parting, break up and separate with someone like parents, friends, lover, etc?

Yeah. But when it is written as 「分かれる」, it's a different story.

Can I also use 別れる when referring to parting with animals, like dog, etc?

Definitely. You can also use it with place words like "town", "country", etc.

Do you mind providing me with a few examples for my better understanding? The examples I found were all regarding lover.

TBH, I feel like saying "Why don't you form some sentences yourself? I'll be happy to correct them."

「明日日本へ行く。３年間は帰国しないつもりだ。家族 や友人たちと別れるのはつらいが・・・。」

「犬のトムと別れてから１年がたった。トム、今頃天国 で楽しく暮らしているかなあ。」

「今日ブラジルへ向かう移民船が出る。いよいよ日本と もお別れだ。」

Thanks, masaegu.