The FS parking lot is gated in the winter. I have seen folks pulled into the entrance in the past in the spring. In many years its stays locked after the snow has substantially melted. Its opening is primarily linked with when Lake of the Crowds opens.
The plowing of this lot has come up in the past. Its a US fee lot and appears to be the only one that I am aware of that has a plowed road in front of it that is not plowed year round. I expect the cog lot is so much more convenient that the FS has elected not to plow the official lot as long as the cog is willing to allow winter hikers to use the cog lots. The cogs approach to hiker parking year round has varied with the owners and year to year. For several years they actively managed hiker parking and charged for it but it sounds as though this has decreased in past years.
The plowing of the base station road may also be one of the reasons the cog allows winter parking. The base station road is at least partially a state road and for many years the state plowing stopped at the Bretton Woods property boundary. The cog wanted to continue the plowing to the base station but the state would not allow it. The reported reason by the state was that the road was not built to be a year round road and that plowing it in the winter and winter use would deteriorate it. This is partially valid as a road that is not plowed in the winter is far less subject to freeze thaw cycles as the snow blanket acts to insulate it. (It is far more questionable with a road that is used for snowmachines as they compact the snow into ice and delay thawing). I expect the far more practical reason was that the road was a popular snowmachine trail and I expect the use of state resources to plow it for its entire length also factored in. The state did allow the Cog to privately plow the Mt Clinton road which also is seasonal road and as most who travel it will acknowledge is in very deteriorated condition. There was also an attempt around in the late eighties by the cog owners to run spring ski trains to the top of Mt Washington. They actively were advertising and selling tickets one season before the USFS put a stop to it and I suspect that that factored into keeping the road unplowed.
At some point in the late eighties/early nineties during the Savings and Loan Crisis the Bretton Woods ski area and the Mt Washington Hotel become pawns in the real estate market leading to at least one maybe two foreclosures. During the turmoil, the ski area got separated from the hotel. The cog owners, who predated the S&L crisis, at one point picked up both properties at foreclosure and one of their goals was to integrate the Cog properties on Mt Washington with the ski area. They had acquired political clout over the years and the state highway departments objections were overruled and the snowmachine route relocated off the road. There was an new attempt for a few seasons to add Cog serviced skiing of Mt Washington "restricted to the Cog property" to the ski areas attractions but a combination of poor snow and high costs doomed this effort. Somewhere around this time, the cog owners who had acquired both the hotel and ski area sold out to the current owners at a reportedly very high profit so the link with the ski area was far less important.
Where I am not sure is what entity assumed the plowing of the road. I believe it is the state but it may be the cog. Based on the width plowed and overall approach to plowing it looks like state plowing as a private contractor would traditionally plow a much skinnier plowed surface. If it is the state, then it would be reasonable that hiker use would be accommodated and I expect that the cog would much rather deal with winter and early spring hiker parking then to plow the road. Despite the cog road being plowed, the FS lot has not been plowed and has remained gated. Over the years there have been some complaints and generally the FS has responded that they lack the resources to plow the lots they do plow and expanding the effort is not possible. Given the preferable option of free parking at the cog versus fee parking at the FS lot I expect the FS realizes that the winter revenue will not cover the plowing costs unless the Cog outright bans hiker parking.