Get node ID that field belongs, in field.tpl.php being displayed in a block on any page
Found my solution!
I was trying to access the node id from the field.tpl.php template because I wanted to add a CSS ID (using the node id) to the field outer wrapper (a div). Which was in this field template.

What I ended up doing was removing this div wrapper from the field template, and just adding the CSS id to the div wrapper in the node.tpl.php template.

Since I only needed this CSS id in certain a view mode, I created a node template for my specific view mode.