介護休業給付金 is the Japanese law allowing nursing care leave for dependents who need it. A quick Google with those kanji can find useful details in Japanese and notably, Employers must provide the option to employees who meet the requirements. The insurance/payments are handled via Hello Work. This is similar to the Maternity, Childcare, etc leaves that grant you some financial support during the process. There are other details as well.

But my question is how this benefit can be used for those of us who are foreigners and our parents might require assistance back in our home countries. Has anyone attempted this? If so, could you give me any insights on the process?

From my investigation (as I do have a parent who requires full-time assistance), I cannot find any restriction for foreigners working in Japan – no exclusion for the dependent residing abroad.

On the MHLW FAQ page for this leave scheme, it does list the necessary documents to support the application:

https://www.mhlw.go.jp/stf/seisakunitsuite/bunya/0000158665.html

3 is 住民票記載事項証明書等（介護対象家族の方の氏名、申請者本人との続柄、性別、生年月日等が確認できる書類）

What documentation could be used in this case, since a Juminhyo and Koseki wouldn't be plausible? I know some countries provide family registries (like Indonesia).