A Royal Commission is an exercise of executive power, and can only be established by letters patent issued by the Governor-General (or the Governor/Administrator/Executive in the case of a state/territory). The Governor-General acts on the advice of their ministers, through the Federal Executive Council (or state/territory equivalent), which is the official constitutional body through which ministers request the Governor-General to sign off on many executive actions. By Westminster convention, the Governor-General is bound to accept the advice of their ministers and can only reject that advice if they identify an extremely serious reason to do so - they do not have any real discretion in their decision otherwise.
Generally, the minister who will countersign the letters patent will be the Prime Minister (or Premier/Chief Minister) but ultimately like most significant decisions of government it is a decision of the Cabinet collectively. If a minister believes there is a need for a Royal Commission, they would make a proposal to Cabinet, and if Cabinet agrees with them, the Prime Minister will announce the Royal Commission and have the relevant documents prepared and approved.
The Parliament does not call Royal Commissions. However, parliamentary committees can hold inquiries, and like a Royal Commission they do have some ability to summon witnesses and order them to answer questions against their will. In practice, parliament's powers to summon witnesses are used quite differently from a Royal Commissioner's powers.
Rudd's petition calls on the House of Representatives to "support" a Royal Commission - the House does not in itself have the power to establish a Royal Commission, but if the House were to "support" a Royal Commission it would place significant pressure on the Government to do so.
The inquiry that Senator Hanson-Young announced is a parliamentary inquiry, being conducted by the Senate Standing Committee on Environment and Communications (see https://www.aph.gov.au/Parliamentary_Business/Committees/Senate/Environment_and_Communications/Mediadiversity). It is not a Royal Commission and its powers are that of a parliamentary inquiry rather than a Royal Commission.
(On occasion, certain Royal Commissions require additional powers beyond the normal powers of a Royal Commission, e.g. the Child Abuse Royal Commission or the Wood Royal Commission into the NSW Police - in these cases, Parliament may pass a bill that makes special provisions for a particular Royal Commission, but otherwise no bill is necessary as there are standard provisions in the Royal Commissions Act already.)
