The Parcel class has a writeStringArray() method. Couldn't your class iterate over the String[][] and call writeStringArray for each?
Assuming your class has a member:
String[][] data;
in writeToParcel()
parcel.writeInt(data.length);
for( int i=0; i<data.lengh; i++ ){
parcel.writeStringArray(data[i]);
}
in createFromParcel()
int size = parcel.readInt();
data = new String[size][];
for( int i=0; i<size; i++ ){
data[i] = parcel.readStringArray();
}