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Consider this circuit. I'm assuming that the resistors have the same resistance. The same amount of current flows through them so they must be in series. But the voltage across each resistor is also the same so they must be in parallel. So are these resistors connected in series or parallel?
Am I just confusing the definitions?
the voltage across each resistor is also the same
No.
The voltage across the total of both resistors is the same no matter their values (assuming the voltage source can provide sufficient current).
The voltages across each of the two resistors are the same ONLY if the two resistors are of equal value. If not, the voltage across each resistor is proportional to its value, per Ohm's Law.
In general, circuits are about galvanic closed loops. You can select any such loop you want.
Here's a GRANU schematic (generally recognized as not useful):
simulate this circuit – Schematic created using CircuitLab
A galvanic closed loop might be \$R_1\$, \$C_4\$ and \$B_1\$, for example. You can draw a circle over that to form a loop. Another loop might be just \$C_1\$ and \$R_4\$. Another loop might be \$C_2\$, \$R_5\$, \$R_4\$, and \$R_3\$, since those are also all connected and if you ignored all the other parts and wiring would form a loop. There are many loops above.
But there aren't many parallel pairs of parts. \$C_1\$ is in parallel with \$R_4\$, \$C_3\$ is in parallel with \$R_6\$, and \$C_4\$ is in parallel with \$R_7\$. That's because in all three cases there is a loop formed only by those two parts. If you cannot find a loop that involves just the two parts, then they are not in parallel with each other.
Now let's look at a different schematic:
Both resistors have the same voltage across them. But they are not in parallel with each other. Having the same voltage across a part does not mean they are in parallel. And not only do both those resistors have the same voltage across them, but they also have the exact same voltages on both ends, too. Yet they are not in parallel because you cannot draw a loop through just those two resistors. It cannot be done.
If, however, you strung a wire from the top end of one resistor to the top end of the other resistor, then they'd be in parallel.
They are in series because current flows through one and then the other. The fact that the voltages and currents are equal are effects of parallel/series connection but don’t define it.
