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The phrase Mother of Parliaments designates England; also: the British Parliament.

The following explanations are from UK Parliament:

Mother of parliaments

On 18 January 1865 British politician and reformer John Bright gave a speech in Birmingham. The speech was part of a long running campaign that culminated in the Reform Act of 1867 and in it Bright coined the phrase ‘mother of parliaments’ when he said that “England is the mother of parliaments.” While not specifically referring to Westminster, the phrase was reported in The Times the next day and quickly caught on.
Soon [sic] the expression is often applied to the Parliament of the United Kingdom because of the adoption of the Westminster model of parliamentary democracy by many countries of the former British Empire.

But, in fact, John Bright popularised, but did not coin, the phrase Mother of Parliaments.—Cf., in quotation 1 below, an occurrence of this phrase dating back to 1857.—And cf. the following from Fifty Years of British Parliament (Boston (Massachusetts): Little, Brown, and Company, 1926), by the British politician Herbert Henry Asquith (1852-1928) [vol. 2: chapter 36: Political Catchwords: page 255]:

Mother of Parliaments

Mr. Bright (Birmingham, January 18, 1865): “We may be proud of this, that England is the ancient country of Parliaments. We have had here, with scarcely an intermission, Parliaments meeting constantly for six hundred years; and doubtless there was something of a Parliament even before the Conquest. England is the Mother of Parliaments.”
The phrase had already become proverbial before it was used by Mr. Bright. It is a vulgar error to speak of the English Parliament as the “Mother of Parliaments.”

These are, in chronological order, the earliest occurrences of the phrase Mother of Parliaments that I have found:

1-: From a correspondence from Australia, published in the Caledonian Mercury (Edinburgh, Midlothian, Scotland) of Thursday 7^th May 1857 [page 2, column 6]—here, it seems to me that the expression Mother of Parliaments may designate the British Parliament rather than England:

On the 2d of March the Haines ¹ Administration fell under an adverse vote. It was the first Cabinet formed under the new system of responsible government ². The vote of censure arose thus: […] Mr O’Shaughnessy moved, without notice of motion, on the 2d of March, “that […] the House feels bound to express its serious disapprobation of the course pursued by her Majesty’s Government in obtaining a vote for the immigration for 1857 […].” Mr Haines intimated that he would look on this as a vote of censure, an interpretation willingly accepted by both sides of the House. An animated debate ensued, said by the Colonial reporters to have been worthy of the “Mother of Parliaments herself,” 29 voted for, and 23 against, the motion; and next day, the same on which Lord Palmerston was defeated at home, the Prime Minister Haines intimated that he and his Cabinet had resigned.

¹ William Haines (1810-1866) was the Premier of Victoria, Australia, from November 1855 to March 1857.
² Frequently with reference to the colonial territories of the former British Empire, the noun responsible government designates a political system in which governments and individual ministers are drawn from, and accountable, to parliament, especially to the lower house, rather than to a monarch or similar power.

2 & 3-: From transcripts of speeches that John Bright (1811-1889), Member of Parliament for Birmingham (Warwickshire, England) from 1857 to 1885, delivered at Birmingham Town Hall:

2-: On Wednesday 18^th January 1865—transcript published in The Birmingham Daily Post (Birmingham, Warwickshire, England) of Thursday 19^th January 1865 [page 5, column 1]:

We are proud of our country; and there are many things in it which as far as men may loyally be proud we may be proud of. We may be proud of this, that England is the ancient country of Parliaments. We have had here, with scarcely an intermission, Parliaments meeting constantly for 600 years; and doubtless there was something of a Parliament even before the Conquest. England is the mother of Parliaments.

3-: On Monday 27^th August 1866—transcript published in The Leeds Mercury (Leeds, Yorkshire, England) of Tuesday 28^th August 1866 [page 8, column 4]:

What is it we are coming to in this country? Why, that which is being rapidly accepted in almost all parts of the world is being persistently and obstinately refused here in England, the home of freedom, the mother of Parliaments. (Applause.) And yet in this England 5,000,000 of grown men, representing more than twenty millions of our population, are to be permanently denied that which makes the only difference between despotism and freedom the world over. (Applause.)

4-: From The Morning Herald (London, England) of Tuesday 22^nd January 1867 [page 4, column 3]:

A question has just been debated in the Prussian Chamber of Deputies which is of far more than merely local interest. It concerns all nations who already enjoy, or who are thinking of adopting, representative institutions. Is it desirable that members of Parliament should receive a fixed remuneration for attendance to their political duties? […]
Are we, whose country is the “mother of Parliaments,” and who would therefore seem to have some right to be heard upon the subject, to denounce a practice so widely differing from our own? Or are we, following the advice of a scarcely friendly prelate, on whose doubtful laudations of liberty we recently commended, to “take a lesson from our own daughters”?

5-: From a transcript of a speech on the extension of the vote to the working class, delivered on Tuesday 22^nd January 1867 by Arthur J. Otway (1822-1912), Member of Parliament for Chatham—transcript published in The Sun (London, England) of Wednesday 23^rd January 1867 [page 1, column 5]:

He firmly believed that with a reformed Parliament all the subjects which required attention would be grappled with, and the House of Commons, invigorated by a large draught of the popular element, would make England, the “mother of parliaments,” an example to her children of prosperity, happiness, and liberty for ages to come.