Is the 边～边 the synonym for 一边～一边?
[边 (verb A) 边 (verb B)] is the shortened form of  [一边 (verb A) 一边(verb B)]. 

Both mean (verb A) and (verb B) are being carried out concurrently.

If (verb A) and (verb B) are single words or very short verb phrases, it is better to use [边 ~ 边~]. For example: "边走边吃" (short and direct)

If (verb A) and (verb B) are long verb phrases, then using [一边~ 一边~] is better. For example "一边工作謀生, 一边照顧他生病的父母" 

(Add a comma to break a long sentence in half for easier reading)

"边工作謀生边照顧他生病的父母" is grammatically acceptable, but it make the sentence too long and doesn't emphasize the two verb phrases individually like using 一边 does.