First, the Bill of Rights applies to Parliament under section 3(a):
This Bill of Rights applies only to acts done— (a)By the legislative, executive, or judicial branches of the government of New Zealand[.]
Of course, any breach cannot be raised in the courts because of parliamentary privilege under the Bill of Rights 1688:
Freedom of speech—That the freedome of speech and debates or proceedings in Parlyament ought not to be impeached or questioned in any court or place out of Parlyament.
In my view, this means Parliament has the obligation to act in a Bill of Rights-consistent manner but is accountable only to itself for breaches of it. (As an aside, there is an interesting question about whether complaints can be made to international human rights bodies. There's others better placed to explore this and it's not really a realistic option here.)
Secondly, there is the question of whether the present rules breach the Bill of Rights. Prima facie the rules infringe section 14:
s14.Freedom of expression— Everyone has the right to freedom of expression, including the freedom to seek, receive, and impart information and opinions of any kind in any form.
However, the prima facies infringement may be permissible if it is a justified limit under section 5:
s5. Justified limitations— Subject to section 4 of this Bill of Rights, the rights and freedoms contained in this Bill of Rights may be subject only to such reasonable limits prescribed by law as can be demonstrably justified in a free and democratic society.
This involves an evaluation of whether the limitation is necessary, suitable, and appropriate. As the Court of Appeal in Moonen v Film and Literature Board of Review  2 NZLR 9 said:
In determining whether an abrogation or limitation of a right or freedom can be justified in terms of s 5, it is desirable first to identify the objective which the legislature was endeavouring to achieve by the provision in question. The importance and significance of that objective must then be assessed. The way in which the objective is statutorily achieved must be in reasonable proportion to the importance of the objective. A sledgehammer should not be used to crack a nut. The means used must also have a rational relationship with the objective, and in achieving the objective there must be as little interference as possible with the right or freedom affected. Furthermore, the limitation involved must be justifiable in the light of the objective. Of necessity value judgments will be involved. In this case it is the value to society of freedom of expression, against the value society places on protecting children and young persons from exploitation for sexual purposes, and on protecting society generally, or sections of it, from being exposed to the various kinds of conduct referred to in s 3 of the Act. Ultimately, whether the limitation in issue can or cannot be demonstrably justified in a free and democratic society is a matter of judgment which the Court is obliged to make on behalf of the society which it serves and after considering all the issues which may have a bearing on the individual case, whether they be social, legal, moral, economic, administrative, ethical or otherwise.
I don't have time to set out an extended analysis but I have strong doubts about whether this rule passes this analysis. The Speaker's remarks in the DomPost suggest the objective of the rule:
Rules covering the filming of members were in place to help viewers see Parliament as it was most of the time. "If that happened, the viewer would see that the behaviour of most members, most of the time, is consistent with the standards of conduct the public expects from MPs," she said. "Unfortunately, by concentrating on gestures such as that by Mr Mark, they leave the viewer with a very poor opinion of all members."
This must be regarded as being a weak objective. It might be just enough to justify a limitation but is not strong. (Presumably, the rule also seeks to stop other MPs grandstanding by waving props etc while another member is speaking.) However, in the light of the doubtful strength of the objective, the limitation would seem to be disproportionate. The prohibition only partially achieves the objective: print and radio being able to convey, albeit in more limited form, the activity which Parliament is seeking suppress. And, most significantly, the prohibition has a detrimental effect on democracy by attempting to hide events which may be of interest and significant to the public. A member's behaviour in the House or other non-speaking activities which they undertake is often newsworthy in its own right.
In my view, it's time for the liberalisation of this rule.
PS I'll try later this week to obtain a copy of the rules and any supporting material which may have touched on these matters.