Fair warning: This answer contains speculative numbers, and may be off by orders of magnitude.
It's just possible, but unlikely.
The issue isn't necessarily with whether or not quantum computers will be able to "parallelize" to that extent. The problem is one of simple physics, one that even quantum computers can't realistically get around. Put simply, there's a limited number of calculations that can ever be performed. This was answered by Thomas Pornin at Security.SE, and I quote some of his answer here:
Let's look at a more mundane perspective. It seems fair to assume that, with existing technology, each elementary operation must somehow imply the switching of at least one logic gate. The switching power of a single CMOS gate is about C*V2 where C is the gate load capacitance, and V is the voltage at which the gate operates. As of 2011, a very high-end gate will be able to run with a voltage of 0.5 V and a load capacitance of a few femtofarads ("femto" meaning "10-15"). This leads to a minimal energy consumption per operation of no less than, say, 10-15 J. The current total world energy consumption is around 500 EJ (5*1020 J) per year (or so says this article). Assuming that the total energy production of the Earth is diverted to a single computation for ten years, we get a limit of 5*1036, which is close to 2122.
Then you have to take into account technological advances. Given the current trend on ecological concerns and the peak oil, the total energy production should not increase much in the years to come (say no more than a factor of 2 until year 2040 -- already an ecologist's nightmare). On the other hand, there is technological progress in the design of integrated circuits. Moore's law states that you can fit twice as many transistors on a given chip surface every two years. A very optimistic view is that this doubling of the number of transistor can be done at constant energy consumption, which would translate to halving the energy cost of an elementary operation every two years. This would lead to a grand total of 2138 in year 2040 -- and this is for a single ten-year-long computation which mobilizes all the resources of the entire planet.
That is the absolute maximum number of elementary operations that can possibly be done. Now let's look at how many chess positions there are...
Let's do some quick numbers. Each of the 64 squares can be empty or hold one of 12 different pieces (R, K, B, Q, K, and P in black and white), so the total number of positions that you could set is at most
1364 = 196053476430761073330659760423566015424403280004115787589590963842248961.
That is about 2 x 1071 different positions. Of course this is a huge overestimate, because most positions are fake (we should eliminate positions with three or more kings, nine or more white pawns, pawns in the eighth rank, quadruple checks, etc). Let's take the square root:
1332 = 442779263776840698304313192148785281,
or about 5 x 1035. By taking the square root we are pretending that for each legal position there is a chess Universe worth of distinct fake positions. This is probably an underestimate, so the true answer must be somewhere in between these two numbers. Now we can confidently say that computers cannot study every legal position in a reasonable time. Even the "tiny" 1332 is too large...
That smallest number ends up being somewhere around 2120 or so.
Let's assume that we represent our boards with a 64-byte string. (Practically it would be handled a little differently, but let's go with it for now.) If I'm remembering my math correctly, a quantum computer would be able to represent this with an 8-byte string, or 64 bits. This leaves us with a total of 2126 to 2130 elementary operations just to store each legal and possible position.
Look at that for a moment. We're not doing anything useful with the information, we're just storing it. And to do so we are mobilizing the resources of the entire planet. Never mind where the storage is physically located. Ignore the whole issue of cooling. Set aside the issue of data transmission. We are diverting enough power to illuminate the Moon just to store the positions.
At the most optimistic of expectations, a quantum computer might be able to solve chess, at the cost of the entire planet's resources. Realistically, that will not happen.