When defining a part, you can get the name of the document (with or without the suffix) into the DESIGNATION field (or any other field) of the Part definition window.
To do this, you have to use the following keywords:

• DOCUMENT_NAME: get the name of the document, including the file extension.
• DOCUMENT_NAME_WITHOUT_SUFFIX: get the name of the document, without the file extension.
  These keywords must be enclosed in the characters [< and >]

Example: [<DOCUMENT_NAME_WITHOUT_SUFFIX>]