$z = 3 + i(2t - 1), t \in [0,1) \\ z = 3 - 6(t-1) + i, t \in [1,2) \\ z = -3 + i(1 - 2(t-2)), t \in [2,3) \\ z = 6(t-3) - 3 - i, t \in [3,4]$
I parameterized a rectangle with vertices at (-3,-i),(-3,i),(3,i), and (3,-i) in the above manner. However, I am having odd results later on in the problem I am working on and I am wondering if it is stemming from a faulty parameterized. Is this valid, and even if it is, is there an even simpler way that I could do it?
You have indeed described a path around a rectangle. As $t$ ranges from $0$ to $4$, the point $z(t)$ moves from $3-i$ to $3+i$ to $-3+i$ to $-3-i$ finally arriving back at $3-i$. The motion is piecewise linear.
If you write it as $$z(t)=\begin{cases}tz_0+(1-t)z_1,& 0\leq t \leq1\\ t'z_1+(1-t')z_2,&0\leq t'\leq1&(t'=t-1)\\ t''z_2+(1-t'')z_3,&0\leq t'\leq1&(t''=t'-1)\\ t'''z_3+(1-t''')z_0,&0\leq t''\leq1&(t'''=t''-1) \end{cases}$$
then it is immediately clear how the point progresses around the path (here the $z_k$ are the vertices of the path).
