Dear fellow Garden Stater,
I definitely believe that locals should have a loud voice about their Park.  I live on the border of the Pinelands National Reserve.  Sadly, I think not many locals even know of its national status (including the Mullica River National Waterway).  Our Pines still get treated with the neglected status of "barrens".  If this fragile ecosystem, one-trillion gallons of pure water, and historically rich land is ever to get the protection it deserves, then the locals have to speak up first. 
Let's turn your Yosemite story around and bring the Californian to the Pinelands.  Can you imagine that?  Can you imagine anybody visiting Yosemite saying, "The Pinelands are next on my 'must-see' bucket list?"  The answers to those questions are profound and will either lead us to the voices we need, or doom us to be over-run with the ever encroaching megopolis.

 I think they are called "National Parks" for a reason-- they belong to all Americans equally even though the locals might be a little more protective -- that's OK!!! LOL

Personally, I don`t think they should have more then anyone else from anywhere U.S.A. after all, the parks belong to all of us equally.

I would actually expect locals to automatically have a strong interest in the parks in their neighborhoods and voice their concerns or suggestions. If they don't, others will be heard.

Something about "the greater good" argument that is often used but in escense it has the effect of redistribution.  The locals have their very economic existance and way of life inalterably changed.  Pretty easy for those that don't have that kind of skin in the game to sit in judgement of those that do.  The reduction of individual rights and responsibilities is running rampant under cover of the "greater good" control argument.  I do enjoy the Parks, often more so than what is experienced by the masses that drive through, it would seem. Should the masses that spend 40 minutes driving through a NP have more sway than an "individual" that spends a lifetime being an integral part of the environment and community?

Inasmuch as locals have the right to be concerned about how a nearby park affects their local economy, they should have a large say in what goes on around their backyard, and they can, through elections, polls, and internet feedback.  Inasmuch as a national park unit is a site of broader significance, the decision making process is left open to a broader decision making process, involving federal departments that can be regulated from elected representatives in congress.  The members of congress most likely to have concerns about the unit would be the local representative or senator.  All in all it seems that there is a nice kind of balance in place to ensure that local voices are heard, just as they should be.  That said, another important thing to remember is that the National Park system started a new concept for the entire world; what locals have in their backyard is not only important to them, but it important to the vitality of a nation and planet.  This is to say, the locals are the most affected by the unit, and should be represented accordingly, but they also do not have a monopoly on the land which has been preserved for all to appreciate and be enriched by, not just some controlling corporate body or interest group.

Whether or not local communities *should* have more influence or not, the history of most national parks that I know shows that those communities *have* always had outsized influence.  
So many national parks have, for instance, become national parks largely because of activism by local (or at least regional) interests.  In the case of the Smokies, of course, Asheville & Knoxville leaders had huge roles (perhaps to the chagrin of other, conflicting, "local" interests in many cases).  Same story for Shenandoah, Blue Ridge Parkway, Cape Lookout, and De Soto National Memorial -- all of which I have studied in depth.  
The nature of that "local" interest does reflect the politics and social/economic/political power structures of any particular region.  The "local" communities (however defined) in many cases may not agree about what course of action to take vis-a-vis a park.  Local community business and tourism leaders, for instance, might well favor the advent of a park they think would bring revenue/business/travelers, while other local constituencies (landowners, often) might oppose a park's actions that they perceive to encroach upon their property.  
Balancing all of these, often competing, local interests against "national" prerogatives, and a general notion of the "public interest" is--and always has been--one of the central conundrums of park management everywhere.  

I think it depends on "who are the stakeholders?"

That why when there was a public comment period of the Point Reyes NS oyster farm issue, there were meetings set up in locations in the San Francisco Bay Area as far as Berkeley. The park visitation is almost exclusively local. The regular customers of the oyster farm are exclusively in Northern California. I think in that case the stakeholders are primarily local.

I think the same goes for the dog walking issues at Golden Gate NRA. It receives a good share of visitors from out of state, but the main visitation are locals going for a walk or taking the dog out.

Now there are parks that have a national clientele, whether it's Yellowstone, Grand Canyon, Yosemite, or the National Mall.

Even when a resource is clearly national, it gets more complex. There was the issue of the cell phone tower in Gardiner, MT outside of the Yellowstone NP boundaries. NPS objected because it could be within view of Roosevelt Arch. However, it was a real town with real people. Who is NPS to tell people that they're not allowed to get better mobile phone service in their own community. Many of the public complaints were about the signals going into the park and ruining other visitors' peace and quiet as people yap on their phones. However, those visitors go back to their homes where they get solid cellular service, and in the meanwhile a real community lives with poor cellular service.