How do I define trait bounds on an associated type?

Yes, you can do that with a where clause:

fn parse<T: Iterator>(mut args: T) -> Result<String, String>
where 
    <T as Iterator>::Item: ToString,
{
   // ....
}

Or, since it's unambiguous which Item is meant here, the bound can just be:

where T::Item: ToString

You can use the Item = syntax:

fn parse<I: ToString, T: Iterator<Item = I>>(mut args: T) -> Result<String, String>

That allows you to simplify this further with the impl syntax:

fn parse<T: Iterator<Item = impl ToString>>(mut args: T) -> Result<String, String>

and finally:

fn parse(mut args: impl Iterator<Item = impl ToString>) -> Result<String, String>

I would consider this a more readable alternative.

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