As is often the case, the answer is Yes and No. Specifically yes in some ways, and no in others.
The chief place where parliamentary systems win out over the separately elected executive branch (at least in terms of preventing a government shutdown) is that the government (i.e. executive) is chosen by the legislative body. This means there cannot be a long-running dispute between the legislative body and the government. If the government loses the confidence of the legislative body, then either a new government is chosen or an election is called. However the US government shutdown was not primarily because of a dispute between executive and legislative branches, but between the two parts of the legislative branch. A parliamentary system would protect against a US shutdown caused by presidential veto of a budget from a Congress that didn't have enough votes to override it - but that's not what happened here (it did happen in 2018).
A bicameral parliamentary system can suffer the same problem. The two houses can disagree about a budget and nothing gets passed. This was theoretically the case with the British parliamentary system until the People's Budget of 1910 (although in practice no dispute had arisen for 300 years). After that processes were modified so the upper house could not prevent the lower house from passing budget legislation. Most parliamentary systems take this approach now, but it is not an inherent property of the parliamentary system.
Anti-deficiency acts are rare in parliamentary systems precisely because the government (executive) are the same people as the majority of the legislative branch. If they want to overspend their budget they can simply pass legislation to authorize it. Without a compelling reason, most places would rather not force a government shutdown just because the legislature can't agree, so acts like that don't get passed. For this reason most governments can continue to spend money even when their budget period has technically ended. The government is going to answer at the polls for their behaviour, so this isn't as much freedom as it sounds. Even if the previous budget has expired, the government is permitted to keep spending enough to keep the government running. (In Canada this is through something called a "Governor General's Special Warrant". Other countries have similar systems.)
Finally, the likely period between failing to get a budget passed and a new budget being passed is:
- a week or so while the goernment tries to negotiate itself out of this mess, and if it can't, goes through the formal process of calling an election;
- a couple of months for the election
- a month or so while the new government sorts out its priorities, appoints new ministers and puts a new budget before parliament
- Another few weeks while the budget is debated
Technically the old government is in charge during the first two of those, but they are trying to get re-elected so everything is just in 'ticking over' mode during that time, They also won't do anything crazy for the same reason.
- Alignment of executive and legislative branch prevents some kinds of deadlock
- Most parliaments have mechanism to prevent deadlock between chambers, but it's not inherent to the parliamentary system
- 'continuations' typically last a few months
EDIT: In response to LateralFractal, most government systems don't have a 'debt ceiling' like the US does - if the expenditure is authorized then raising the money (borrowing) is also authorized. That's not specific to a parliamentary system.
ANOTHER EDIT: Some comments ask about the government in "caretaker mode". In a Westminster style parliament there is no "caretaker mode". When an election is called the government remains with all the same powers until the election results are known and the prime minister either continues with a new mandate or resigns.