Powerful Parity Pyramid!

Here are 7 different (non-symmetrical) solutions that I found using the Python program below:

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# Powerful Parity Pyramid
# Python

import string
digs = string.digits + string.ascii_letters

# function that changes a decimal to a different base
def int2base(x, base):
    if x < 0:
        sign = -1
    elif x == 0:
        return digs[0]
    else:
        sign = 1
    x *= sign
    digits = []
    while x:
        digits.append(digs[int(x % base)])
        x = int(x / base)
    if sign < 0:
        digits.append('-')
    digits.reverse()
    return ''.join(digits)

# function that gets the next line in the pyramid
def nextline(s):
    l = ""
    for a in range(0, len(s) - 1):
        p = s[a:a + 2]
        if p == "01" or p == "10":
            l += "1"
        else:
            l += "0"
    return l

# function that finds the number of all odds in the pyramid
def pyroddsum(s):
    st = s
    for a in range(0, len(s) - 1):
        s = nextline(s)
        st += s
    return st.count("1")

# go through each possible combination of odds and evens
#  of the bottom line, and print out the ones that make
#  a total of 14 odd numbers in the whole pyramid
for a in range(0, 2 ** 7):
    seq = int2base(a, 2).zfill(7)
    if pyroddsum(seq) == 14:
        print(seq)

I will refer to the nth picture from the left, of the mth row from the top, as [m,n].

Take such a pyramid as that described in the question and resize each of its cells into equal-sized squares. Rotate each square by ninety degrees and your pyramid will end up looking something like [2,1]. In fact, arrange your pyramid like [2,1]; rotate it until its originally-top corner is at the bottom left. Each of the figures in the image represent this new representation of the type of pyramid described in the question. The figures of the first row consist of extensions 'down' the pyramid, such that squares are formed. Red squares represent even numbers, while blue, odd.

Confirm that [2,1] is a valid pyramid, that it fits the description of a pyramid given in the question. Then confirm that [2,2] is a valid pyramid. Then confirm that [1,2] is valid. By induction, [2,3] is a valid pyramid, and all such pyramids extended from [2,3] are valid. This is because they all will consist of already-validated pyramids. (I will leave out rigorous write-ups of inductions, because they are more succinctly imagined than verbally articulated.)

In each figure in row 2, flip the parity of the rightmost squares, the 'bottom row's of the pyramids they represent. This gives the figures of row 3. Flipping the parities of the bottom-most row of a valid pyramid keeps the pyramid valid. Confirm this by flipping the bottom-most row in each of the only four possible 3-cell pyramids. Therefore all pyramids in row 3 are valid.

Confirm that [3,1] is evenly split between even numbers and odd numbers. Locate the instances of [3,1] in [3,2] that are along the 'bottom row' of [3,2]. These portions of [3,2] may be ignored because we already know they are evenly split between even and odd numbers. Is the remaining portion of [3,2] evenly split between evens and odds? This portion does resemble [1,2]. Confirm that [1,2] is evenly split between evens and odds. So [3,2] is evenly split between evens and odds. Therefore all pyramids in row 3 are both valid and evenly split between evens and odds. This is because they all comprise of previously confirmed and validated pyramids.

Note how many rows are in each pyramid in row 3: it starts with three rows, and repeatedly, four rows are appended. This shows that pyramids with numbers of rows, $n$ , such that $n\equiv3\mod4$ , are fillable with an equal number of evens as odds, which is what the question is asking about. $7\equiv3\mod4$ implies that the answer to the question is 'Yes'.

We can very quickly explore some of those 'fun questions' as well. To obtain [4,2], append a valid row of numbers to [3,2]. To obtain [4,1], append a valid row of numbers to [3,1]. Do these such that [4,2] and [4,1] each have the same number of evens as odds. The slant edge of [4,3] may comprise of a concatenation those of [4,1] and [4,2]. This is because of the pattern apparent in the transition from [3,2] to [3,3]; [3,3], recall, is made of [3,2], [1,2], and [3,1]. All other parts of [4,3] are instances of [1,2], which we've already confirmed evenly partitions evenly into evens and odds. Therefore [4,3] is a valid pyramid and has the same number of evens as odds. All figures in row 4 are build in a similar way and by induction, all figures in row 4 are valid and each have the same number of evens as odds.

Note how many rows are in each pyramid in row 4: it starts with four rows, and repeatedly, four rows are gained. This shows that pyramids with numbers of rows, $n$ , such that $n\equiv4\equiv0\mod4$ , are fillable with an equal number of evens as odds, which is what the question is asking about.

Finally, how many rows must such a pyramid as that described in the question have in order to contain an even number of cells? The number of cells in such a pyramid are represented by triangle numbers, any and all. The formula for the nth triangle number is $\frac{n(n+1)}2$ . Any even triangle number is formed by the formula with an $n$ that satisfies either $n\equiv0\mod4$ or $n\equiv3\mod4$ . These are exactly the sets of numbers of rows that we've confirmed can produce valid pyramids with the same number of evens as odds.

We now know, at least intuitively (as I left out rigorous writing), that all such pyramids as described in the question, all those with an even number of cells, can be filled with the same number of evens as odds.