直接翻成He persuaded their dad out of drinking.
没有必要加succeed in因为persuade意思是：成功说服,当中已经包含了“成功”这层含义,如：
The salesman persuaded us to buy his product.
那个推销员成功说服了我们买他的产品.
此外,这句话还可以有一下几种表达方式：
①He persuaded their dad to quit drinking.
②He talked their dad out of drinking.
talk sb into doing/talk sb out of doing 这两种表达也表示说服某人做（不做）某事