Are all statutory instruments in the United Kingdom subordinate to an Act passed by Parliament?
In a strictly technical sense, yes - but only because a statutory instrument is by definition legislation issued under a power granted by Act of Parliament.
Specifically, the Statutory Instruments Act 1946 defines an SI as follows:
Where by this Act ... power to make, confirm or approve orders, rules, regulations or other subordinate legislation is conferred on His Majesty in Council or on any Minister of the Crown then ... any document by which that power is exercised shall be known as a “statutory instrument” and the provisions of this Act shall apply thereto accordingly.
(Some text omitted.)
However, if the question is also seeking legislation of any kind which is not subordinate to an Act, then as Dale M's answer suggests, the government (acting in the name of the Monarch) is able to use powers under the Royal Prerogative. In particular:
An Order in Council made under the Royal Prerogative is primary legislation, and does not depend on any statute for its authority, although an Act of Parliament may change this. This type has become less common with the passage of time, as statutes encroach on areas which used to form part of the Royal Prerogative.
Matters which still fall within the Royal Prerogative, and hence are regulated by (Prerogative) Orders in Council, include dealing with servants of the Crown, such as the standing orders for civil servants, appointing heads of Crown corporations, governance of British Overseas Territories, making appointments in the Church of England and dealing with international relations.
Returning to the question:
If so, is the intention that the thrust of law is democratically accountable, but for expediency the details can be passed with less democratic oversight?
It could argued that it's more about convenience than expedience - though under some circumstances, SIs can come into force before Parliament has had a chance to review them.
In any case, they tend to be used to specify details (e.g. regulations) that would be too cumbersome to be included in an Act, and which might change over time. They are also commonly used to bring Acts, or parts of Acts, into force on particular dates (commencement orders).
Also, the power to issue SIs is not limited to the government. Acts of Parliament may also delegate this power, as appropriate, to the UK's devolved governments, local authorities, and other official institutions.
IIUC EU directives are often implemented using statutory instruments and the “negative resolution procedure”, but these are presumably subordinate to an Act passed in 1973(?) when we acceded to the EC?
Yes; it's the European Communities Act 1972. See also this question relating to how EU directives become law in the UK.