As per my understanding, you cannot crawl SharePoint 2013 from SharePoint 2010 neither publish SharePoint 2010 Search Service Application to SharePoint 2013. Both will not work due to search schema difference of both versions. However, you can do the reverse by following these steps.
- Set Up Trust across Farms
- Setup Search and Publishing
- Set Permissions
- Connect to the Published service application
- Associate Service application with the Web Application
- Configure Server-to-Server authentication between farms
- Configure Content Source
This is source of these steps.
In the 7th step of above article, you need to put SharePoint 2010 sites URLs in SharePoint 2013 content source and run a Full Crawl. Before you start a Full Crawl, make sure your content crawling account of SharePoint 2013 Search Service has Full Read permissions on SharePoint 2010 sites.
In this way, you'll crawl SharePoint 2010 sites from SharePoint 2013 and then can publish the SharePoint 2013 Search Service Application to SharePoint 2010 environment to show the same search results in both environments.
Hope it helps.