시간이 없어서 이제 테스트해 봤습니다.
원인은 Parameter의 개수가 문제 더군요.
Parameter 추가 전에 Clear를 해주시면 잘 됩니다.

qryParam->Parameters->Clear(); //추가

[수정 소스]
int PCount;

AnsiString szSQL = "SELECT STUDENT_ID, STUDENT_NAME "
                            "FROM TB_CLASS "
                            "WHERE CLASS_ID =:parmClassID "
                            "ORDER BY STUDENT_ID ";

if(qryParam->Active) qryParam->Close();
qryParam->SQL->Clear();
qryParam->Parameters->Clear(); //추가
qryParam->SQL->Add(szSQL);
qryParam->Parameters->AddParameter();
PCount = qryParam->Parameters->Count - 1;

qryParam->Parameters->Items[PCount]->Name = "parmClassID"; 
qryParam->Parameters->Items[PCount]->DataType = ftString;
qryParam->Parameters->Items[PCount]->Direction = pdInput;
qryParam->Parameters->Items[PCount]->Value = "A-1";
qryParam->ParamCheck = true;
qryParam->Open();