I have an experience of three and a half years and I have never heard of a petition as stated by you as a "Direction Petition". According to my limited knowledge, an Application seeking directions from the Hon'ble Court can always be filed in the concerned Court where the case is pending adjudication. In case of decided matters the Application may be filed in the name of Miscellaneous Application.
An Application for directions can also filed before the District Judge( in civil matters) and Session's Judge( in criminal matters ) seeking the direction upon the subordinate judges to expedite the process. Suc Applications shall only be allowed in cases where there is an apparent delay in serving justice and only on the proof of bonafide grounds by the Applicant. Only the parties to the case can file such applications for direction.
Another remedy could be by way of filing a Writ Petition before the High Court seeking direction upon the subordinate courts to expedite the process.
Thirdly, a writ petition under Article 226 may be filed by any aggrieved person seeking direction pertaining to a particular subject and case on the infringement of a legal right and a fundamental right as provided by the Constitution of India.
Further, a petition may be filed before the Hon'ble Supreme Court under Article 32 only if any fundamental right is infringed excluding the legal rights.
You may feel free to contact for any further clarrifications.
Tarveen Singh Nanda