My father is a money saver and it seems that he likes to see his remaining balance in his bank account grow.
ご質問ありがとうございます。
"My father is a money saver"＝「私の父は貯金家です。」
（又は、"my father likes to save money"＝「私の父はお金を貯めるのが好きです」という言い方も出来ます。）
"and it seems that he likes to see his remaining balance in his bank account grow."＝「そして彼は自分の銀行口座の預金残高が増えるのを見るのが好きなようです。」
ご参考に。
ーMy father is good at saving money.
「父は貯金が得意だ」＝「貯金家だ」
to same money で「お金を貯金する」
ーMy father is a real saver.
「父はすごい貯金家だ」
saver で「貯蓄家」
例：
My father is a real saver. He loves looking at his bank balance going up, up, up.
「父はすごい貯蓄家で、銀行の預金残高がどんどん上がっていくのを見るのが好きだ」
bank balance で「銀行の預金残高」
ご参考まで！
アメリカ合衆国 
カナダ 
