No, not directly through your website with standard Forums installed.
Your member gets that because he has "Yes" selected for "Notify me of replies:" in his user profile preferences - entirely his choice.
Alternatively he can set that to "No" and selectively check the box when he makes a post, or by clicking "Watch this topic for replies" at the bottom left of a topic he is viewing. All of this is basically Forums101.
Server specs (Server OS / Apache / MySQL / PHP / DragonflyCMS):