  Models that produce Axion-Like-Particles (ALP) after cosmological inflation
due to spontaneous $U(1)$ symmetry breaking also produce cosmic string
networks. Those axionic strings lose energy through gravitational wave emission
during the whole cosmological history, generating a stochastic background of
gravitational waves that spans many decades in frequency. We can therefore
constrain the axion decay constant and axion mass from limits on the
gravitational wave spectrum and compatibility with dark matter abundance as
well as dark radiation. We derive such limits from analyzing the most recent
NANOGrav data from Pulsar Timing Arrays (PTA). The limits are similar to the
$N_{\rm eff}$ bounds on dark radiation for ALP masses $m_a \lesssim 10^{-22}$
eV. On the other hand, for heavy ALPs with $m_a\gtrsim 0.1$ GeV and $N_{\rm
DW}\neq 1$, new regions of parameter space can be probed by PTA data due to the
dominant Domain-Wall contribution to the gravitational wave background.
