While I am not an expert on French Constitutional Law, I first of all would not think that the fact that France joined NATO under a prior constitution would matter, because a constitution governs how a government is allowed to act now. The whole point is that it is the presently effective set of principles governing how laws and treaties are made in the country. Moreover, as @ruffle notes in the comments, the rule was essentially the same in the 4th Republic.
The harder question is whether an invocation of Article 13 of the NATO treaty amounts to a new treaty that must be approved by parliament, or is merely part of the course of performance of an existing and already approved treaty, which does not.
The language of Article 13 could be more clear although comparison with Article 11 helps explain it:
This Treaty shall be ratified and its provisions carried out by the
Parties in accordance with their respective constitutional processes.
The instruments of ratification shall be deposited as soon as possible
with the Government of the United States of America, which will notify
all the other signatories of each deposit. The Treaty shall enter into
force between the States which have ratified it as soon as the
ratifications of the majority of the signatories, including the
ratifications of Belgium, Canada, France, Luxembourg, the Netherlands,
the United Kingdom and the United States, have been deposited and
shall come into effect with respect to other States on the date of the
deposit of their ratifications. . . .
After the Treaty has been in force for twenty years, any Party may
cease to be a Party one year after its notice of denunciation has been
given to the Government of the United States of America, which will
inform the Governments of the other Parties of the deposit of each
notice of denunciation.
The main obligation under the NATO Treaty is Article 5 which states:
The Parties agree that an armed attack against one or more of them in
Europe or North America shall be considered an attack against them all
and consequently they agree that, if such an armed attack occurs, each
of them, in exercise of the right of individual or collective
self-defence recognised by Article 51 of the Charter of the United
Nations, will assist the Party or Parties so attacked by taking
forthwith, individually and in concert with the other Parties, such
action as it deems necessary, including the use of armed force, to
restore and maintain the security of the North Atlantic area.
Any such armed attack and all measures taken as a result thereof shall
immediately be reported to the Security Council. Such measures shall
be terminated when the Security Council has taken the measures
necessary to restore and maintain international peace and security.
Part of the purpose of Article 53 of the French Constitution seems to be that since a Treaty trumps a domestic law under the French Constitution under Article 55 of the French Constitution (unlike U.S. law under which a domestic law may override a treaty), that the legislature should have to consent to treaties that have the effect of modifying or limiting domestic law.
A denunciation under Article 13 of the NATO Treaty relieves the French State of the duty to treat war on one NATO members as a war on all NATO members, but doesn't deprive France of the ability to affirmatively declare war in such circumstances, so that doesn't seem to do much harm to the principle of French legislative autonomy.
But, a denunciation under Article 13 of the NATO Treaty also deprives the French State of the benefit of protection from other NATO members which is something that the legislature cannot obtain on its own if this denunciation is given effect.
Ultimately, I think that the most convincing argument is that Article 13 grants the right to withdraw to a "Party" and not to a particular individual in the political process associated with that party such as a Head of State. This says to me that its must be a collective legal action of the state and not merely the action of an official of the state carrying out the terms of the Treaty.
So, I am inclined to think that parliament's approval would be required.
Fortunately, the one year waiting period of Article 13 of the NATO Treaty ought to be sufficient to give the Constitutional Council the power to rule on the validity of a unilateral denunciation pursuant to Article 54 of the French Constitution, and to order the President to withdraw France's denunciation before it becomes effective if a unilateral denunciation is found to be improper.