The no-hair theorem of General Relativity ensures that, under some conditions, black holes are completely characterized by only three quantities: their mass, electric charge, and spin. This means, in particular, that a black hole does not "remember" how it was formed. For example, if you have two stars, one made of matter and the other of antimatter, and both collapse into a black hole each, the final black holes are undistinguishable if they have the same mass, charge, and spin. Even though they were originally made of different "materials".
Now, Hawking radiation does not depend on the details of the collapse, only on the final state of the black hole. As a consequence, both of those black holes will emit neutrinos and antineutrinos in equal amounts, and they will with all other neutral particles. Hence, you do have, e.g., violation of baryon number conservation.
In fact, let me quote the Wikipedia page for baryon number conservation:
The conservation of baryon number is not consistent with the physics of black hole evaporation via Hawking radiation. It is expected in general that quantum gravitational effects violate the conservation of all charges associated to global symmetries. The violation of conservation of baryon number led John Archibald Wheeler to speculate on a principle of mutability for all physical properties.
The reason for the expectancy that all global symmetries should be violated is, in general lines, due to the fact that the black holes can only be characterized by "long-range properties". Mass, charge, and spin have long-range effects (by means of gravity, electromagnetism, and gravity again), so their conservation can be "seen from afar". In particular, from outside the black hole. However, baryon number is a global charge, with no consequences perceivable at a distance. To know the baryon number of a star, you'd have to actually go there and count the baryons, you can't use something like Gauss's law (you can do that with mass and spin). Hence, there's no way for the outside of the black hole to perceive the baryon charge, and it is just "forgotten". The symmetry is indeed violated.
It is important to recall that the Standard Model is formulated in the absence of gravitational interactions, and there are some properties that hold in flat spacetime, but not necessarily will generalize for curved spacetime. Hence, it is important to be careful when trying to apply familiar results to a vastly different situation.
Going more straight to your question, while I've never seen anyone discussing the case for neutrinos in particular, I see no reason for why black hole evaporation would have to behave in a way that keeps the number of neutrinos from increasing. In other words, the only "violations" I can see happening in this scenario are of global symmetries, which is expected.
Particles are not a fundamental concept
I noticed that it might be worth adding this comment in here: particles are by no means a fundamental or observer-independent concept in Physics. It should be recalled that Particle Physics and Hawking radiation are described in terms of Quantum Field Theory, which is a theory of fields, not particles. This becomes extremely proeminent in Quantum Field Theory in Curved Spacetimes, because we learn that different observers do have different notions of particles, and this is precisely what lies behind the Hawking effect.
I wrote a bit more about this in connection with the Hawking effect in this answer.
Disclaimer on the No-Hair Theorem and on the Information "Paradox"
As pointed out by safesphere in the comments, the first paragraph of this answer can be the subject of some discussion. Whether or not astrophysical black holes remember how they are formed is a discussion still happening in the scientific community related to the so-called Black Hole Information "Paradox". I discuss a bit more on what this "paradox" is and what are the existing views in this answer. My use of quotation marks might hint at my personal view, which also influences the choice of wording of the first paragraph.
Hence, I'm stretching the No-Hair Theorem in that paragraph. In fact, the theorem is called a theorem by physicists, but due to the heavy assumptions that go into the current formulation it is often called a conjecture by mathematicians.
Someone with a different view of the Information "Paradox" could argue that somewhere in the evaporation process the Physics will not be well-described by Quantum Field Theory in Curved Spacetime and the Universe will find a way to bring back the information lost within the black hole. Nevertheless, as far as I know, even then people are perfectly willing to let go of global conservation laws (see, e.g., this post).
