You can set permissions on individual records.  The issue is that if the user uses the Infoblox Grid Manager GUI to update the record, they will need read only access to all parents of the record to browse to it.  The user would need read only access to the DNS View, the DNS Zone, any subzones if applicable, and finally Read/Write access to the record.

You might be able to get around that by using global search at the top right.  By searching, you're not navigating through the rest of the objects to get to the DNS record.

Oh, and if default permissions exist, you can negate them on the DNS record by setting permissions to DENY for the users/groups that would otherwise be inherited.