From the formula for a geometric series, \[p(x) = 1 + x^2 + x^4 + x^6 + \dots + x^{22} = \frac{x^{24} - 1}{x^2 - 1}.\]Likewise, \[q(x) = 1 + x + x^2 + x^3 + \dots + x^{11} = \frac{x^{12} - 1}{x - 1}.\]At first, it may look like we can write $ p(x) $ as a multiple of $ q(x) $: \[\frac{x^{24} - 1}{x^2 - 1} = \frac{x^{12} - 1}{x - 1} \cdot \frac{x^{12} + 1}{x + 1}.\]Unfortunately, $ \frac{x^{12} + 1}{x + 1} $ is not a polynomial. A polynomial of the form $ x^n + 1 $ is a multiple of $ x + 1 $ only when $ n $ is odd. So, we can try to get close by considering $ \frac{x^{11} + 1}{x + 1} $. Let's also multiply this by $ x, $ so that we get a polynomial of degree 12. Thus, \begin{align*} \frac{x^{12} - 1}{x - 1} \cdot \frac{x(x^{11} + 1)}{x + 1} &= \frac{x^{12} - 1}{x - 1} \cdot \frac{x^{12} + x}{x + 1} \\ &= \frac{x^{12} - 1}{x^2 - 1} \cdot (x^{12} + x) \\ &= (x^{10} + x^8 + x^6 + x^4 + x^2 + 1)(x^{12} + x) \\ &= x^{22} + x^{20} + x^{18} + x^{16} + x^{14} + x^{12} + x^{11} + x^9 + x^7 + x^5 + x^3 + x.\end{align*}This is a multiple of $ q(x) $ that's very close to $ p(x) $. In fact, when we take the difference, we get \begin{align*} &p(x) - (x^{22} + x^{20} + x^{18} + x^{16} + x^{14} + x^{12} + x^{11} + x^9 + x^7 + x^5 + x^3 + x) \\ &\quad = -x^{11} + x^{10} - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1.\end{align*}Now, if we add $ q(x), $ we get \begin{align*} &p(x) + q(x) - (x^{22} + x^{20} + x^{18} + x^{16} + x^{14} + x^{12} + x^{11} + x^9 + x^7 + x^5 + x^3 + x) \\ &\quad = 2x^{10} + 2x^8 + 2x^6 + 2x^4 + 2x^2 + 2.\end{align*}We can also write this as \begin{align*} &p(x) - (x^{22} + x^{20} + x^{18} + x^{16} + x^{14} + x^{12} + x^{11} + x^9 + x^7 + x^5 + x^3 + x - q(x)) \\ &\quad = 2x^{10} + 2x^8 + 2x^6 + 2x^4 + 2x^2 + 2.\end{align*}So, we took $ p(x), $ subtracted \[x^{22} + x^{20} + x^{18} + x^{16} + x^{14} + x^{12} + x^{11} + x^9 + x^7 + x^5 + x^3 + x - q(x),\]which we know is a multiple of $ q(x), $ and ended up with $ \boxed{2x^{10} + 2x^8 + 2x^6 + 2x^4 + 2x^2 + 2} $. Since the degree of this polynomial is less than the degree of $ q(x), $ this is our remainder.