yesterday he called his father using his mobile phone

可以。但是句意完全不同。
yesterday he called his father using his mobile phone 意思是昨天他用手机给爸爸打电话。

yesterday he called his father to use his mobile phone 意思是昨天为了用一下手机，他给爸爸打电话。

ing（using）形式表示补充说明。做状语。to use 没有这种用法。

用现在分词做状语，补充说明，我刚才特意上网查到的。

不可以阿， 语法错了：
yesterday he called his father using his mobile phone
昨天他用手机给爸爸打了个电话
如果说：yesterday he called his father to (tell him to)
use his mobile phone (to call him back) 意思就变成：他给爸爸打电话请爸爸用手机回复他
第一个动词 （过去太） he called 和 第二个动词连接要用 present participle=动词加-ing 才可以

应该要换成to use 吧！！只有call sb to do sth不是吗？？如果要用using,应该改成yesterday he called his father ,using his mobile phone 意思是昨天他用手机给爸爸打电话。

不可以
原句得意思是，他昨天用他的手机给爸爸打了电话。
换成to use 的话，就变成，他为了用手机而给爸爸打了电话

可以吧