It would seem clear, from various legal precedents, that the government has the power to restrict the Constitutional rights of convicted felons, for at least the time that they are under parole.
A random example:
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Colorado’s Supreme Court ruled on Monday that parolees—felons released from jail and serving their mandatory period of parole—cannot vote until their parole is complete. The unanimous ruling stated that parole constitutes a portion of the “full term of imprisonment,” the constitutional phrase in question, and is therefore not a violation of state law.
Having the power to do this doesn't mean it's a good idea, of course. The point I am trying to make is that, if the courts agree that felons under parole can be denied the right to vote (which, I would argue, is at least as fundamental a right as the right to bear arms), it can certainly restrict their ownership of firearms. A felon with a gun presents a danger, obviously; a felon who can vote does not. The disenfranchisement of felons is strictly for the purpose of punishment; the disarming of felons has the desired practical effect of reducing future crimes.
An interesting question arises once the felon has gone through the entire period of parole. Some states remove the right to vote
forever.
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In 13 states a felony offense can result in the loss of voting rights even after the sentence has been completed, and often for life.
I don't agree with such a policy, but the point is that some states have been doing this without, apparently, successful Constitutional challenge. If they can do this, it would seem logical that they could also prevent a felon from ever legally owning a firearm.
We can argue whether this is a good idea or not, depending on the crime involved.