The direct effects of a presidential pardon depend on the state law where the pardoned person lives, not federal law.
While a presidential pardon will restore whatever rights may have been lost due to the offense in question, it does not eliminate or expunge the record of said offenses. If a party inquires into such an offense, the individual in question is required to disclose said information.
The ability to hold political office is determined on a state-by-state basis. If a pardoned individual did want to run for office after being declared a felon, the state would have confer that right to them if it had been removed.
Eligibility to run for office is also dependent on whether the pardon is full, partial, absolute or conditional.