could you please tell me what is “by their Father” means

In the halachos of kashrus we find that the mixing of milk and meat can either by done by the milk and meat mixing.
Or by their taste mixing The taste of food can transfer either when the food is at the heat of 45 Celsius (while some are machmir at 40 Celsius) or if the food is sharp and cut with a knife. if the food is not sharp or hot the taste which is in the food will not transfer over via the vessels which are being used unless the vessel being used is unclean as the hen there is actual food left there which is giving over the taste.
Therefore, theoretically speaking if you wash down the mixer after usage you can surely use it for dairy and thereafter for pareve with no concern at all.
For this reason there is no problem to use the same machine for both dairy and pareve yet if the machine is not totally clean you should wipe it down the machine after using for dairy before you use it for pareve if you want the pareve to certainly remain pareve.