or series (which breaks the connection to phones placed after the one unhooked in the series.
I got what you are speaking about (and why you call connection "serial").
Even in this scenario, the phones has been connected in parallel on A & B wire of phone loop. But line wires has been passed thru phone to next phone unless disconnected by cradle switch (while off-hook).
For the purpose of your question, you can consider both cases equivalent.
if I can replace my external incoming line to the first jack with the SPA Line 1 connection, and have the rest of the phones connected, or if I need to run an additional wire per phone.
As I mentioned above, you can consider both cases equivalent. Additional "per phone" wire will change nothing. It will work, or will not work (but for other reason).
Possible reasons for "it doesn't work":
1. if phones are "line powered" then multiple phones in parallel may draw so much current from line; as a result, line will be considered seized (off-hook, busy) all the time - overall loop resistance needs to be higher than 10000ohm for unseized state.
2. any phone may not be satisfied with ringing signal (20-25Hz, trapezoid) thus the particular phone may not ring, even if other phones will ring (or no phone will ring if all phones are affected)
3. requirements of your phones exceeds 5 RENs supplied by SPA112 - some or all phones may not ring reliably or at all, SPA112 SLIC will be overloaded
4. off-hook voltage and current may not be enough for particular line-powered phone (some or all phones may be affected; note - even wiring length and wire diameter affects this) - off-hook current supplied is 18–25 mA, voltage seen by phone depends on both line and phone resistance
5. other reason I didn't remembered now
But to make my answer clear - you need not to run additional wires.
