That all sounds fine to me:
(The following steps might happen in a different order but the end result will be the same)
Step 1: The solicitor pays for his parking. The VAT element ('X') is now in NCP's bank.
Step 2: NCP pay their VAT bill to HMRC. X is now with the tax man.
Step 3: Your firm reclaims the VAT. X is now back with your firm.
Step 4: Your firm invoices the client, including the VAT ('Y') on the car parking disbursement. (X = Y)
Step 5: Your firm pays it's VAT bill. Y is sent to HMRC.
The end result is that your company 'breaks even' but the client pays the VAT to HMRC. (As you've said, HMRC gets the VAT twice, in steps 2 & 5, but pays it back to you in Step 3, so the tax man only receives one VAT payment in total).
Chris