You can use the LogUnsubEvent code found here to unsubscribe the person from the Business Unit and also log the event in the tracking associated with the email send job.
In order to use this correctly, you need to use the MicrositeURL(if using a Landing Page) or CloudPagesURL (if using a CloudPage) function in your email unsubscribe link. This ensures that the subscriber attributes are accessible within the Landing Page/Cloud Page
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SET @sid = emailaddr
SET @jid = jobid
SET @listid = listid
SET @batchid = batchid
SET @reason = "Profile Center Unsubscribe"
SET @lue = CreateObject("ExecuteRequest")
SetObjectProperty(@lue,"Name","LogUnsubEvent")
SET @lue_prop = CreateObject("APIProperty")
SetObjectProperty(@lue_prop, "Name", "SubscriberKey")
SetObjectProperty(@lue_prop, "Value", @sid)
AddObjectArrayItem(@lue, "Parameters", @lue_prop)
SET @lue_prop = CreateObject("APIProperty")
SetObjectProperty(@lue_prop, "Name", "JobID")
SetObjectProperty(@lue_prop, "Value", @jid)
AddObjectArrayItem(@lue, "Parameters", @lue_prop)
SET @lue_prop = CreateObject("APIProperty")
SetObjectProperty(@lue_prop, "Name", "ListID")
SetObjectProperty(@lue_prop, "Value", @listid)
AddObjectArrayItem(@lue, "Parameters", @lue_prop)
SET @lue_prop = CreateObject("APIProperty")
SetObjectProperty(@lue_prop, "Name", "BatchID")
SetObjectProperty(@lue_prop, "Value", @batchid)
AddObjectArrayItem(@lue, "Parameters", @lue_prop)
SET @lue_prop = CreateObject("APIProperty")
SetObjectProperty(@lue_prop, "Name", "Reason")
SetObjectProperty(@lue_prop, "Value", @reason)
AddObjectArrayItem(@lue, "Parameters", @lue_prop)
SET @lue_statusCode = InvokeExecute(@lue, @overallStatus, @requestId)
SET @Response = Row(@lue_statusCode, 1)
SET @Status = Field(@Response,"StatusMessage")
SET @Error = Field(@Response,"ErrorCode")
]%%