Mryntu

There's a big machlokes in shas about ridesharing. Namely, who pays for it: the mother (parent) or the child?

"עובר ירך אמו" vs "עובר לאו ירך אמו"

"Uber should be charged to the parent" vs "Uber should NOT be charged to the parent"

Thoughts on how we might pasken?

Would the halacha be similar or different for Lyft?

I believe that the one who pays is the one who does the action of making the arrangements, as we hold עובר לעשייתן, that the Uber goes to the one who does the work for it.

While rosends's answer covers Uber, what about Lyft? Berachos 40a says:

אמר רב חסדא קיתון לפת

Rav Chisda said, a kid on the Lyft.

What does this mean? Does this mean that the kid pays for it, or that the parent pays for the Lyft on top of the kid? On 43a, the Gemara elaborates:

הסבה לפת מהניא

The grandfather [for the] Lyft helps.

This clearly implies that the elder generation pays for it, and so the parent pays for the child.

This question is asked as long ago as the Gemara:
(Sanhedrin 57b):

משום רבי ישמעאל אמרו אף על העוברין מאי טעמיה דרבי ישמעאל דכתיב (בראשית ט, ו) שופך דם האדם באדם דמו ישפך איזהו אדם שהוא באדם הוי אומר זה עובר שבמעי אמו- it was said in the name of rabbi yishmael: why even the Uber users? As it says, ‘the one who pours money of man, his own money will be poured in man’. Who is this ‘man whose will be poured in man’? That who Uber’s in his mother’s stomach”
We see from here, that one who Uber’s in his mother’s stomach, I.e. the son, he ends up pouring his money into man (gives his money to the cabby) [דם in this instance means money, like דמים, not blood]

The achronim talk about this as well:
(shita mekubetzes Bava Basra 139b)

עובר בן הוא לא עדיף- Uber, a son is not preferred
We see then, that the shita mekubetzes argues with the Gemara! The chachmas Adam solves this contradiction:
(chachmas adam 28)

עובר בן ח' או בן ט'- Uber, [only] an 8 or 9 year old son
Only from 8 or 9 years old does a son need to pay for his parents Uber.

The Gemara is talking about a child under the age of 8, who is not expected to finance his parents Uber. The shita mekubetzes is talking about an older child, from 8/9 and up, who is expected to finance his parents Uber.
This is how we pasken (as there are no other sources that talk about it).
As for Lyft, it says in herchev davar al Bereshit (27):

מצות כיבוד אב כמו שאירע ליפת. - it is a mitzva of kibbud av va’em like what happened with Lyft.

We see from here that we learn Hilchos kibbud av va’em for Uber from Lyft! So for sure the Halachos will be the same!