Barbados was able to do so because under Article 49 of Barbados' Constitution, Parliament is empowered to alter the document through the passage of an Act through both of its Houses - as long as that bill is supported by at least two-thirds of the members of each House.
The bill in this case, the Constitution (Amendment) (No. 2) Bill 2021, had its first reading in the House of Assembly on September 28th, was passed by a vote of 25-0 the same day, and passed the Senate on October 6th. The bill allows for the election of a new President, to assume office on November 30th.
Quite why this was decided without a referendum is slightly more difficult to pin down - Prime Minister Mottley, the leader of the Barbados Labour Party (which won every seat in the elected House of Assembly after the last election in 2018) - gave this response when questioned in September 2020:
Williams: And from here on in, you're just going to actually do this - there's no referendum involved?
Mottley: That's right, no, twenty years ago we committed to a referendum, and since then every government, the major political
parties have agreed that this is where the country must go - it is our
judgement and we certainly campaigned on it in our manifesto that
while we committed to a referendum on other issues, we didn't on this
one, and we made it clear that this is where we believe the country
must go. And in fact, there was a group of eminent Barbadians who
would have before the last election, spoken to the former Prime
Minister, spoken to myself as Leader of the Opposition, extracted
commitments from both major political parties, and we are at the stage
now where we're just simply going to - as Nike would say - just do it,
make it happen.
However, I don't see any reference to this point in their 2018 manifesto, although the party does, on page 45, commit to:
- Introducing National Dialogues, National Referenda and consulting
with Barbadians on major national issues, such as the
decriminalisation of recreational marijuana.
There is also a vague mention of modernising the "principal organs of governance as outlined in our Constitution" on page 41, but as far as I can see, no mention of the monarchy, presidents, or a republic.
The Prime Minister also referred to their commitment to a referendum 20 years ago - presumably a reference to a Referendum Bill introduced in late 2000 which did not make it through Parliament before it was dissolved in 2003. In 2005, the Barbados Labour Party committed further to holding a referendum on this issue:
I turn now to the question of the Referendum.
In 1994 as one of the pledges made to the people of Barbados, this
Administration proclaimed that on a matter as fundamental as the
change of our governance, such as would be involved in a move to a
Republican status, the opinion of the people should be specifically
and deliberately canvassed by way of a Referendum.
The entire Cabinet and the wider family of the Barbados Labour Party
are firmly of the view that such a commitment is still fundamentally
valid and must be fully honoured.
Later this year a Bill to amend the constitution to give effect to the
various and sundry recommendations of the Forde Commission will be
introduced into Parliament. The Amendments will make reference to the
President as the Head of State of Barbados.
In advance of any such change, and in keeping with our commitment to
the people, we propose to hold a Referendum specifically on the matter
of the change from the Monarchical to the Republican system of
In pursuit of this, we propose also to introduce into Parliament a new
Referendum Bill that will seek public support for the move, expressly
in the form in which it is recommended in the Forde Constitution
Review Commission’s Report.
I have noted that the Leader of the Opposition, the Honourable Clyde
Mascoll has declared his support in principle for both the change to
Republican status and the holding of a Referendum.
I'm unable to find any manifesto commitment to abolish the monarchy without testing the support of the people in a referendum, nor any public change in their stance of requiring a referendum before enacting such a change prior to the last election in 2018 - however, doing so was fully within the power of the Parliament under the Constitution. It seems that the decision was made that as both major political parties supported the transition to a republic, it was decided that it was politically acceptable or expedient to do so without explicitly canvassing the electorate.
While you mention several countries which held referendums on transitioning to a republic, take the examples of Trinidad and Tobago, the Parliament of which amended the Constitution to abolish the monarchy in 1976 without a referendum, and of Fiji, which declared itself a republic in 1987 after a coup d'état.