Educational attainment is definitely a person level variable.
To find the highest level of educational attainment for persons within a household, you would need to retrieve the PUMS person records and look for the maximum value within all persons within a household.
In SAS, I would sort the person records by the household ID; then use PROC SUMMARY of the data, BY the household ID; and output the MAX value for the SCHL field (or EDUCD if you're retrieving the PUMS data via IPUMS.)
In R, I'd download the PUMS data using the get_pums function in the package "tidycensus". But I haven't mastered R to try and create household level variables (e.g., employed workers in the household; tally of persons age xx-xx in HH; highest educational attainment of persons in HH)
If you were interested in just the educational attainment of the HOUSEHOLDER (1st person record in the household), then I would recommend using the IPUMS SDA system. But person #2 or #3, etc., might indeed have a higher educational attainment level than the householder!!
Chuck