Is what the questioner did—throwing the remaining pebbles all at once for fear of crowding, and sending a rosary to the Grand Mosque and charity to six poor people—an expiation for stoning Jamrat al-Aqabah? And is his intention valid in the second Hajj when he says: "O Allah, if my first Hajj was not accepted by You, then accept this second Hajj from me; and if my Hajj was accepted the first time, then I am performing Hajj on behalf of my parents"?

Throwing the pebbles (ramy al-jamarat) is an obligation (wajib). If one misses three pebbles or more, a compensatory sacrifice (dam) is required. The act of sending charity does not suffice for the blood sacrifice; rather, a sheep must be slaughtered within the sacred precinct (haram) and its meat distributed to its poor. A lost item in the Haram is like a lost item outside it. A rosary (subhah) is considered trivial, so there is no harm in taking it, and one is not required to send a similar item back to the Haram. If you performed the Hajj with its pillars (arkan) and obligations (wajibat) and avoided its nullifiers, then your Hajj is valid, and the obligation is fulfilled. Acceptance is a matter of the unseen. A Muslim is not required to repeat their obligatory Hajj out of fear that it was not accepted. Performing Hajj with a wavering intention counts for the individual performing it, not for someone else.