It's often because you're dealing with the thickening agent over the heat.
A virtually foolproof method is to use 2 tablespoons of cornflour to 1/2 pint of milk.
If you put your milk in a saucepan, add a knob of butter and a bit of pepper then bring it to boiling point.
Remove the pan from the heat at this point.
Have your cornflour ready mixed in a cup - 2 tablespoons let down to a creamy consistency with COLD milk.
It will have solidified whilst you've been boiling your milk, so give it a good stir and then pour it into your pan of just boiled milk, stirring well as you add it.
When it's incorporated, then return the pan to the heat and stir constantly until the sauce has thickened.
It will only take a few seconds for this.
Cook the sauce for around a minute or so then add your chopped parsley and stir through - taste and adjust seasoning before serving.
Hope that helps you get over your lumpy parsley sauce :-)
You can use other herbs or grated cheese in this basic white sauce as well - it's very versatile.