The drag and drop functionality doesn't work with the Parallels client for Windows.
Local drives are not redirected.
1. Check in the connection properties → Local resources if the drives are enabled for redirection.
2. Check if on the RDSH the policy for Do not allow drive redirection is set to Not configured or Disabled:
Computer Configuration → Windows Components → Remote Desktop Services → Remote Desktop Session Host → Device and Resources Redirection → Do not allow drive redirection
Alternatively, you can check it in the System registry:
HKLM\SOFTWARE\Policies\Microsoft\Windows NT\Terminal Services\fDisableCdm
1 means that drive redirection is disabled
0 - it's enabled
If there is no such entry the parameter is not configured.