Linear Chirp parameters from a segment of the signal
Amplitude $A$ is trivial. Square and low-pass filter/average; then take the square root.

Without noise (or with good SNR), you could simply calculate a derivative:

\begin{align}\frac{\partial\, s\}{\partial t} = -A\sin\left(\omega_0t + \frac\alpha2t^2 +\phi_0\right)\cdot\left(\omega_o+ \alpha t\right)
\end{align}

and low-pass filter the result

$$\text{LPF}\left\{\frac{\partial\, s\}{\partial t}\right\} = \omega_o + \alpha t$$

From the resulting linear function, $\alpha$ can directly be extracted as the slope, and knowing the slope, $\omega_0$ is just the constant summand.

With $A$, $\omega_0$ and $\alpha$ known, you can simply compare at any fixed $t$ to find $\phi_0$.