The 2nd paragraph ["I want to be able to select only the parent (without the children) as well as have the children be selectable on their own or all together when clicking the parent"] makes it sound like you want to enable selection of the children checkboxes when the parent is checked, and disable selection of the children checkboxes when the parent is un-checked. That's the pattern you are looking for. Are you looking guidance on how to implement that?
I think part of the confusion is the use of "parent" and "child." This can imply a class-based hierarchy, where the children are specific types of the parent. That seems to be the type of relationship most of the comments are thinking of. (See taxonomy in wikipedia: https://en.wikipedia.org/wiki/Taxonomy)
But parent-child can also imply a containment-based hierarchy, where the children are merely contained (physically or logically) within a parent. HTML is a good example. A img (image) tag can be a child of a p (paragraph) tag...
<img src="images/animals/cat.jpg" />
...without the img tag being a type of the p tag. (See wikipedia meronomy: https://en.wikipedia.org/wiki/Meronomy)
It seems like a meronomy might be the type of parent-child relationship you're referring to.
Could you clarify?
I'm only guessing here, but I suspect a scenario that might be similar to your needs is something like this:
 Yes! Send me your product for only $9.99!
 Include gift wrapping for only $2.50 more
 Send by overnight delivery for only $19.25 more
 Include postal insurance for an additional $12.50
In this case, it would only make logical sense to select the "children" if the parent was selected, but selecting the parent without the "children" is also a logical possibility.