At common law, parliamentary privilege does not extend to the republication of words spoken in Parliament: Stockdale v Hansard (1839) 112 ER 1160. To overcome this, Parliament passed the Parliamentary Papers Act 1840, which provides absolute privilege for a report published “by or under the authority of either House of Parliament” (ie. Hansard).
Section 3 of the Act provides qualified privilege for publishing an “extract or abstract” of Hansard. In relation to statements made under parliamentary privilege in breach of a court order, the Committee on Super-Injunctions wrote in its report on Super-Injunctions, Anonymised Injunctions and Open Justice (May 2011):
In order to obtain the benefit of the immunity from civil or criminal proceedings under section 3, the individual must prove, before a jury, that the relevant material was published in good faith and without malice. It is an open question whether publication of any extract from or abstract of Hansard which had the effect of frustrating a court order and was deliberately intended to do so would be held to be in good faith and without malice …
Where media reports go beyond, or fall outside, the ambit of protection provided by the 1840 Act, it is the common law which determines whether there is any protection from contempt proceedings for breach of court orders. Mr Woolf noted that while Wason v Walter (1868) 4 QB 73 established a limited common law protection in defamation proceedings for honest, fair and accurate reporting of Parliamentary proceedings, there was ‘no reported case which authoritatively decides the extent of protection against proceedings for contempt available in respect of fair and accurate reports of proceedings of Parliament’ …
It therefore appears to be an open question whether, and to what extent, the common law protects media reporting of Parliamentary proceedings where such reporting appears to breach the terms of a court order and is not covered by the protection provided by the 1840 Act. What is clear is that unfettered reporting of Parliamentary proceedings (in apparent breach of court orders) has not been established as a clear right.
In its report on Parliamentary Privilege (June 2013), the Joint Committee on Parliamentary Privilege said that the Parliamentary Papers Act would generally not apply to the contemporary media, and that the defence of qualified privilege was limited to defamation actions:
An “abstract” was defined by the 1999 Joint Committee as a “summary or epitome”. Thus media reports of what goes on in Parliament, even if they draw on documents published by order of the House, such as Hansard, do not generally enjoy the (qualified) protection afforded by section 3 of the 1840 Act—a point confirmed by Sarah McColl, Solicitor Advocate in the BBC Editorial Legal Department, in her oral evidence on behalf of the Media Lawyers Association …
Outside the field of defamation, it does not appear that media reports of parliamentary proceedings (as opposed to extracts or abstracts) enjoy legal protection. This arises most obviously in the case of breaches of court injunctions …
The Joint Committee recommended that the law should be reformed:
On balance, we do not support extending absolute privilege to all reports, including media sketches and summaries, of proceedings in Parliament … we consider that the existing protection of qualified privilege—that is, that all fair and accurate reports are protected, unless they can be proved by the claimant to have been made maliciously—already provides a robust defence of press freedom …
At the same time, we acknowledge that the media need clarity and certainty. The Parliamentary Papers Act 1840 provides qualified protection for all “extracts and abstracts” of parliamentary publications (including broadcasts), but as we have already said these terms do not appear to cover media reports or editorial comment …
We endorse the recommendation of the 1999 Joint Committee that [the] Parliamentary Papers Act 1840 should be replaced by modern statutory provisions … We recommend that these new provisions should …
establish that qualified privilege applies to all fair and accurate reports of parliamentary proceedings in the same way as to abstracts and extracts of those proceedings;
provide that in all court proceedings in respect of such fair and accurate reports, extracts or abstracts, the claimant or prosecution shall be required to prove that the defendant acted maliciously …
However, it appears that Parliament has not taken up the Joint Committee’s recommendation, and the extent of any common law protection remains an open question. Accordingly, the article linked in the question states that “The BBC is not naming Soldier F for legal reasons.”