I know that out of the box a SharePoint 2013 farm cannot consume Service Applications from a 2010 farm.
Are there any third party solutions that enable this? I have tried searching online, but have not been able to find anything.
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Their is no backward compatibility for services application so that you cannt use the 2010 services in the SharePoint 2013.
Reason is simple, SharePoint 2013 have more features then the old versions, when you try to get a data from lower end edition it will throws the error because New version will ask more stuff(features / option) which are not present in lower edition.