During charging or discharging a lead acid battery both the positive and negative electrodes will undergo reduction and oxidation the same time.
For instance during discharging process, the cathode will react with the sulfuric acid and will give the electrolyte electrons i.e. oxidation. And simultaneously the cathode will gain electrons from the circuit i.e. reduction.
So, the cathode and anode undergo both process reduction and oxidation during charging and discharging.
Am I right?
The lead acid battery has two electrodes, one made of metallic lead, and the other made of lead dioxide $\ce{PbO2}$. Remember that, whatever the operation (charge or discharge), the anode is always the electrode where oxidation occurs.
Let's consider first the discharge process.
In the discharge operation, the lead electrode is the negative electrode as it produces electrons. And it is working as an anode, as $\ce{Pb}$ is oxidized from Zero to $+2$ according to $$\ce{Pb + SO4^{2-} -> PbSO4 + 2 e-\tag{1}} $$The $\ce{PbO2}$ electrode is the positive electrode; it is working as a cathode, and the $\ce{Pb}$ atom is reduced from $+4$ to $+2$ according to : $$\ce{PbO2 + 4 H+ + SO4^{2-} + 2 e- -> PbSO4 + 2 H2O \tag{2}}$$ At the end of the discharge, both electrodes are similarly made of $\ce{PbSO4}$.
It is time to consider the charging process. In the charging process, the cell is working as in electrolysis and the preceding equations are reversed.
In the original lead electrode, equation ($1$) is reversed, and $\ce{PbSO4}$ is reduced to $\ce{Pb}$. The electrode is not an anode any more. It is a cathode. But the sign of the electrode is maintained negative, as the lead ion $\ce{Pb^{2+}}$ needs electrons to be regenerated into $\ce{Pb}$. So the minus sign (-) can be printed or engraved on the electrode. It stays the same in the charge and in the discharge processes.
In the second electrode, initially coming from $\ce{PbO2}$, equation (2) is reversed. $\ce{PbSO4}$ is oxidized to $\ce{PbO2}$ again. So this electrode, being oxidized, becomes an anode. But it remains the positive electrode. So the plus sign (+) may be printed on the $\ce{Pb/PbO2}$ plate. This (+) sign stays the same in the charge and in the discharge processes.
