  It was recently proposed that a giant flare of the blazar PKS B1424-418 at
redshift $z=1.522$ is in association with a PeV-energy neutrino event detected
by IceCube. Based on this association we here suggest that the flight time
difference between the PeV neutrino and gamma-ray photons from blazar flares
can be used to constrain the violations of equivalence principle (EP) and the
Lorentz invariance for neutrinos. From the calculated Shapiro delay due to
clusters or superclusters in the nearby universe, we find that violation of the
equivalence principle for neutrinos and photons is constrained to an accuracy
of at least $10^{-5}$, which is two orders of magnitude tighter than the
constraint placed by MeV neutrinos from supernova 1987A. Lorentz invariance
violation (LIV) arises in various quantum-gravity theories, which predicts an
energy-dependent velocity of propagation in vacuum for photons and neutrinos.
We find that the association of the PeV neutrino with the gamma-ray outburst
set limits on the energy scale of possible LIV to $> 0.01 E_{pl}$ for linear
LIV models and $>6\times 10^{-8} E_{pl}$ for quadratic order LIV models, where
$E_{pl}$ is the Planck energy scale. These are the most stringent constraints
on neutrino LIV for subluminal neutrinos.
