Do switches allow parallel connections?

You should note that CSMA/CD is all but obsolete. It was used with (repeater) hubs and half-duplex links and only for 10 and 100 Mbit/s. Modern networks are switched throughout and use full duplex. Gigabit speed is practically considered a given.

In packet-switched networks, all data is broken down into packets which are then transferred from source to destination individually. While each physical link can only transfer one of these packets at any given time, the time period in question for a packet is in the range of microseconds (a maximum sized Ethernet frame over a 1 Gbit/s link takes little more than 12 µs).

The granularity of the packetized data practically allows each link to be multiplexed for any data that leaves or enters a network port, so a single port can be used for a large number of applications "simultaneously" (in the scope of human perception).

If I want to send data from B to A and also at the same time from A to C. Is it possible to transfer data in such a way simultaneously?

Yes. With a full-duplex link, this can literally happen simultaneously. On an obsolete half-duplex link, one of those transfers on A's port would be slightly delayed by CSMA/CD, but unless the link is congested that delay is far below anything a human could notice.

Will it cause collision?

Only on a half-duplex link and if "simultaneously" means within the same time window required to transmit a frame. (CSMA/CD tries to avoid collisions by the "carrier sense" part; collisions only happen when transmission attempts start very close to each other in time. If the first transmission's carrier is sensed by the other station wanting to transmit, the latter will wait until the channel is free again.)

Will managed switches work differently than unmanaged switches in this case?

Managed or not doesn't make a difference. The basic forwarding function is exactly the same.

Unmanaged switches just learn MAC addresses and forward frames based on the learned addresses - that's it.

Managed switches give you far greater control over a network, including priorization, partitioning/segmenting, link aggregation, user/node authentication, filtering, diagnostics, ...