As noted in the variable description for PARSE, this 9-digit identifier is based on the official Swedish six-digit code for county, municipality, and parish.
The Swedish National Archives document from my previous post states:
The 9-digit geographical code described above is also used in the Swedish historical censuses from 1880, 1890, 1900 and 1910 that have been delivered to the North Atlantic Population Project (NAPP) (https://www.nappdata.org) hosted by the Minnesota Population Center (MPC) at the University of Minnesota. The 9-digit code is used to identify parish of residence and birth parish in the Swedish censuses. This enables a researcher to map the content in the censuses to the shape-files described in this document.
The NAPP data mentioned in the document are now included in the IPUMS International database; based on the document, I think that kommuner actually stands for municipality rather than parish. If you see the “Ref” row of the G_units_Names table, it notes that columns 1-2 of the 9-digit SCB codes are the county, columns 3-4 are the municipality codes, columns 5-6 are for uniquely identifying parishes, and the final three columns differentiate between different versions of the parish. I cannot speak to the Digitala gränser codes (though I only saw 4 digits in the files available there, which makes me think these are municipality codes rather than parish codes), but hope this information helps you assess their comparability with the IPUMS codes.