No, held the EAT in Mardner v Gardner.
The Claimant succeeded in her case before the employment tribunal and applied for costs. Despite the costs threshold having been crossed, one of the reasons the employment tribunal refused to award the Claimant her costs was that she was not personally out of pocket because her case was funded by her legal expenses insurers.
Applying Parry v Cleaver  AC 1, the EAT held this was an error because public policy dictates that the paying party should not avoid the costs consequences of their unreasonable conduct merely because the receiving party prudently entered into an insurance contract. That would allow the paying party to appropriate the benefit of the receiving party's contract, which would be wrong.