There are a few ways to handle this, but they tend to not belong in a requirements specification or a design specification. This is especially true if you are building the software internally, but need to release your interface documentation to a broader audience. You probably don't want to release all of your software requirements and internal design information to external entities interested in your interfaces.
At work, we use interface requirements specifications (or interface requirements documents) and interface control specifications (or interface control documents). The interface requirements specifications tend to only exist at a system or subsystem level. By the time you are dealing with a software-only component, the interface requirements have been defined, so we'll just include the pertinent information from the higher level specification (either inclusion or trace - both are easy with a requirements management tool).
The advantage of this is that we tend to not release component specifications (requirements or design) to external parties. Software requirements specifications contain requirements levied by the business to support long-term technology development goals as well as internal customers from systems engineering, SQA, manufacturing, and others. These are requirements that we may not want to expose to external customers for a number of reasons (especially constraints or requirements in support of long-term tech development).
The interface control specification or interface control document simply provides a specification for the inputs and outputs between two or more components, subsystems, or systems. How we write these varies. For some products, a component outputs a standard format so we have an ICD for that format and a separate ICD for the input. We can then go and assemble these pieces in different ways for customers, making a sort of volume of documents all linked together.
Something else to keep in mind, a document doesn't necessarily need to be a Microsoft Word or PDF document. A wiki page or collection of wiki pages (with appropriate edit control and revision history, especially with the ability to export to a static format) is just as much a document as other formats. It's more about the content being available and appropriately linked together in a meaningful manner than the way it's ultimately pulled together.
Interface design and control is a practice from Systems Engineering. NASA has an interesting document about the subject - "Training Manual for Elements of Interface Definition and Control". However, depending on your field, it's probably overkill. You may be able to learn from these practices and tailor them to an appropriate level.