The Supreme Court ruled that parents are entitled to a preliminary
injunction against that policy while the lawsuit continues. That is:
the Board should be ordered to notify [parents] in advance whenever one of the books in question or any other similar book is to
be used in any way and to allow them to have their children excused
from that instruction.
Exercising this right obviously requires that parents be notified when
classroom instruction includes such material, so that they can take
their children out of that class.
No right is established by the present ruling other than the plaintiffs' right to the preliminary injunction they requested.  However, in coming to that decision, the SC did conclude that the plaintiffs are likely to ultimately prevail on the merits of their case, and in doing so it rejected some lower-court analyses that would have made that outcome less likely.
Supposing that the plaintiffs do prevail, that will establish that their existing first amendment right to freedom of religion precludes their school district from using the materials in question in their curriculum without affording the plaintiffs and similarly-believing parents an opportunity to opt their children out, or providing some other suitable accommodation.
But I wonder how broad this is. Is
it specific to LBGTQ+ material, or does it apply to anything the
parents might object to on religious grounds?
This is answered for the current decision at the end of its syllabus:
Thus, the petitioners have shown that they are entitled to a preliminary injunction. Specifically, until all appellate review in this case is
completed, the Board should be ordered to notify the petitioners in advance whenever one of the books in question or any other similar book
is to be used in any way and to allow them to have their children excused from that instruction.
That is, the injunction is specific to circumstances involving the particular books and similar ones, where it appears "similar" should be taken to mean promoting the same or similar opinions and moral judgements about gender identity, same-sex marriage, and associated issues.  The injunction granted does not extend to instruction involving other kinds of materials, even if the paintiffs have similar religious objections about them.
Moreover, the court considered the particular nature of the materials in question, among other details of the case, in determining that a preliminary injunction was warranted.  It did not explicitly rule out other religious objections to instructional materials being sufficient grounds to support a claim for infringement of religious freedom, but it did not leave the impression that it would have considered arbitrary other religious objections to be sufficient.
I suppose you're interested in what the larger implications would be of the plaintiffs prevailing in the underlying case, but even if we suppose that they do ultimately prevail, there's no way to predict that before the decision is published.
Since teachers may not
be able to predict what some parents would object to, does this
essentially require them to distribute their entire lesson plan to
parents?
No.
- The injunction is limited to the particular books at issue and similar ones.  There is no reason to think that it would leave teachers uncertain about which lessons require parent notification and provision for opt out.  There may well be other lessons and classroom materials that elicit objections on religious grounds, but they are not covered by the injunction. 
- The district originally did provide an opt-out arrangement with respect to the materials in question, pursuant, apparently, to established district guidelines for respecting religious diversity.  It was discontinuation of that arrangement that occasioned the lawsuit. The injunction seems to require the district to restore the particular opt-out program they previously provided, which, as far as I can tell, did not involve teachers distributing their entire lesson plans, nor anything substantially equivalent. 
- Although I don't expect it is within individual teachers' discretion, the district likely has the alternative of settling the entire lawsuit by agreeing to permanently remove the materials in question from their curriculum.  That would relieve teachers from any kind of notification requirements related to the suit. Inasmuch as the district claims that among their reasons for discontinuing their opt-out program in the first place was a large and increasing number opt outs, it seems like that might have been overall well received by their community if they had handled it that way in the first place. That equation is different now that the issue has been so politicized.