A few corrigenda and then some tl;dr information for you.
The short version is that several nasty civil wars were fought to make sure that this sentence of yours is legally and politically false:
if a party can manage to push a bad law through Parliament (especially in a majority Gov't situation), and through the Senate (a longer-term and more "sober" house, but political nonetheless), it can be stopped by the head of state
The Parliament of Canada consists of the House of Commons, the Senate and the Queen of Canada. (This is not what you say in your sentence above; you meant "the House of Commons" rather than "Parliament" presumably).
The consolidated Constitution Acts (1867-1982) says:
IV. 17. There shall be One Parliament for Canada, consisting of the Queen, an Upper House styled the Senate, and the House of Commons.
Also:
Yes, it is true that the GG can stomp on a bill, but if it's passed both the House and the Senate and hasn't raised any serious questions, then it's mostly a formality. Likewise, she can't "abolish" a law without the approval of the other parts of government either
is only true with respect to "without the approval of the other parts of [Parliament]", and then only if you read "The Queen" as "the Prime Minister", effectively.
The Queen (as an institution, rather than as a person) has a vital Constitutional role that is quite active. IV. 54. requires that the Queen make a recommendation before any spending bill may lawfully be passed by Parliament; 55. and 57. require that the Queen give Assent to any bill before it is enacted.
Both Houses of Parliament have their own rules that further restrict their activities with respect to IV. 54. (and the Senate is further constrained by 53. which requires that all money-raising bills originate in the House of Commons). Finally, 56. allows the Queen to disallow any Act within one year of its passage. In general, very little legislation may lawfully be passed without both the Queen's *prior and subsequent* consent.
In practice the Queen of Canada is read as the government of the day, since she may not exercise her powers directly or through the Governor-General without the advice of the Prime Minister (which is often delegated to other ministers and officials).
The Recommendation requirement and the statutory and house rules allow even a minority government to control all budget legislation. A House of Commons and Senate dominated by opponents cannot increase taxes *or* spending without the approval of the government (i.e. the Prime Minister of the day); if they want to do that, they must force the government to resign. (As examples, they can do this either by refusing to pass any legislation at all, or by starving the government of Supply, or the House of Commons can indicate that it has no confidence in the government's ability to govern).
Non-budget legislation can be passed by both Houses of Parliament in spite of the government's wishes, in which case the government likely would refuse to give the bill Royal Assent, killing it dead.
A government that controls Parliament and in its last few months starts passing Acts that the immediate successor government strongly disagrees with could see the use of the Disallowance power used against those Acts by the incoming government.
The power of the Queen of Canada to act contrary to the advice of the Prime Minister was fully extinguished in the wake of the King-Byng affair (1925), the Balfour Declaration (1926), the Statute of Westminster (1931) and finally by section 12 of the consolidated Constitution Act which ultimately finalized Parliament's abolition of the Queen of Canada's ability to exercise any personally held (or delegated) constitutional powers without the advice of the Prime Minister (or the Cabinet Committee and Emergency Committes of the Privy Council of Canada where the Prime Minister is incapacitated).
Moreover, no British (or Canadian) monarch has exercised the powers above without the explicit advice of the (equivalent of) the Prime Minister since Queen Anne. Several substantial civil wars, and several smaller ones, in England in the 1600s finally eliminated the monarch's right to use his or her constitutional powers against a government.
In Canada the King-Byng Affair was the Viscount Byng of Vimy, the then Governor-General of Canada, acting on orders from the government of the United Kingdom against the government of Canada. That was won by Canada, and rapidly led to the dismemberment of the British Empire.
The only other unilateral action by a Governor-General of Canada was when Sir Charles Tupper refused to resign his government in the aftermath of an election that Tupper conclusively lost. Although the then Governor-General (who acted on orders from the government of the UK in London) was able to force Tupper to resign, it led to a further clarification of the undisputed right of a Prime Minister (of Canada, or of the UK) to confront and seek the confidence of (their respective) two sequential Houses of Commons. If he or she loses the confidence of the current House of Commons, he or she may direct the monarch to dissolve Parliament and issue writs for a general election. If the newly elected House of Commons does not indicate its confidence in the Prime Minister, then the Prime Minister must resign.
This rule is an acceleration of the approach which the Marquess of Aberdeen (the G-G who dismissed Tupper) *should* have followed, namely that the House of Commons should have refused to pass Supply bills and other legislation until Tupper resigned *of his own accord*, which is in line with the pre-Confederation parts of the Canadian Constitution, notably Magna Carta, the Bill of Rights (1689) and the Act of Settlement (1701).
[bad law] can be stopped by the head of state if it isn't seen to benefit Canadian society
An unelected official opposing an elected House of Commons will quickly put that official out of office.
The very first complaint in the of the Bill of Rights (1689), which is still clearly part of the Canadian Constiution is about:
assuming and exercising a power of dispensing with and suspending of laws and the execution of laws without consent of Parliament
the technicality here being that the King (and now Queen) are part of Parliament, however the very first two subsstantive clauses say:
the pretended power of suspending the laws or the execution of laws by regal authority without consent of Parliament is illegal
the pretended power of dispensing with laws or the execution of laws by regal authority, ..., is illegal
The Prime Minister therefore derives his or her authority from the whole of Parliament, and the House of Commons in particular delegates to the Prime Minister the task of representing its consent to the use of the Executive Powers.
The last Monarch to forget this was forced to abdicate. His predecessor who sparked the revolution that led to the Bill of Rights (1689) was beheaded.
If Byng or Aberdeen had not acted on the advice of one of the great officers of state (i.e. Cabinet Ministers) of the United Kingdom, they both likely would have been sacked. As it was elected officials in the UK, the remedy was the rapid end of UK politicians' authority over most of the now-Commonwealth.