I would reckon that if you were acting on information from a telephone or radio call prior to proceeding at an unfamiliar airfield then the airfield is responsible.
If you had just decided without asking and after landing that you would park on the grass for a picnic, then you would be responsible.
The grey area might be if you were advised that you can tie down on the grass, and you lashed a 500kg aircraft to a 5gallon drum of water, which then blew away taking an F16 with it then it may be your fault depending on who or where you got the 5gallon drum from.
Excuse my drivelling.