If all of the numbers are either odd or even this means that the table must be an even number and the shift of the table will be odd if all of the numbers are odd or even if all of the numbers are even, since even+odd=odd and even+even=even.
The table must be an even number since if the table was an odd number then not all the values shown will be all odd or all even numbers. This is because of the way that odd numbers and even numbers multiply. If you multiply an even number by either odd or even both will result in an even number. (Even x Odd = Even, Even x Even = Even) If you multiply an odd number by an even number you get an even number however if you multiply by an odd number you get an odd number back. (Odd x Odd = Odd, Odd x Even = Even) This means that if our table is an even number when we add or subtract our shift we will always get numbers of a single parity whatever we multiply our table by.
For example if our times table is 4 and we shift up by 3 then whatever number we multiply 4 by then add 3 to the product we will always get an odd number since 4 is even and 3 is odd and even+odd is odd.
4x1+3=7 (Odd)
4x2+3=11 (Odd)
4x3+3=15 (Odd)
4x4+3=19 (Odd)
4x5+3=23 (Odd)