First, regarding the crank's bottom bracket recommendation: The crank uses a square taper bottom bracket spindle, and would not be compatible with the Shimano Hollowtech II BB that may be currently in the bike. You could swap out to a BB-UN50 or BB-UN70 from Shimano in a 68x115mm size, to fix the crank/BB compatibility issue if that is the case.
That leaves the tooth difference question. The first thing to check is the tooth capacity of the front derailleur. The bike in question uses a Shimano Tiagra, 10-speed front derailleur.
Shimano has a Product Info website which provides specifications and compatibility information by part type, model number and product series.
The relevant information for a Tiagra Double Front Derailleur is listed under the part number (which is engraved on the back of the actual front derailleur), in this case, the FD-4600 or FD-4700.
The information we are looking for is the maximum tooth difference between large and small chain rings. This is identified as maximum capacity.
As you can see in the image below, the maximum capacity (tooth differential) is 16 teeth.
That means that a 42/26 should just be in the range of use for a Tiagra 10 speed double front derailleur.
The next category to look at is the top gear teeth, which is the recommended range of tooth count for your large ring.
As you can see in the image below, the recommended top gear tooth count for this derailleur is between 46 and 52 teeth. Now, you may be able to set it up and get it to work, but perhaps not as well as it should, or perhaps not at all.
I would conclude that this would require more changes to be compatible with a chain ring set that small. Perhaps a front derailleur, BB and crank in all.
Or you may not be able to place a front derailleur low enough on the frame to be functional off a 26 tooth ring (for a double). I would think it could be worked out, but probably not with these parts.