Gulp task to increment a major, minor, or patch level of the version number of the last build
The first thing that I see is that:

    major = parseInt(versionArray[0]) + 1;


Is a bit redundant. You already say:

major = parseInt(versionArray[0]);


Just do:

major += 1;


(This is similar for all the major, minor and patch statements)

You can also use a switch-case instead of if statements so instead of:

if (ver === 'major') {
    major = parseInt(versionArray[0]) + 1;
}
else if (ver === 'minor') {
    minor = parseInt(versionArray[1]) + 1;
}
else if (ver === 'patch') {
    patch = parseInt(versionArray[2]) + 1;
}


You have:

switch (ver) {
    case 'major':
        major += 1;
        break;
    case 'minor':
        minor += 1;
        break;
    case 'patch':
        patch += 1;
        break;
}