On Parabola it doesn't work out of the box because for some reason the libre kernel blacklists the free tg3 driver (which is required). Broadcom Tigon3 deblob from https://linux-libre.fsfla.org/pub/linux-libre/releases/LATEST-5.3.N/deblob-5.3 removes both the proprietary firmware and the driver from the kernel itself even though for this model, the firmware is not necessary for it to work acording to https://unix.stackexchange.com/questions/336917/tigon-ethernet-firmware-error-message-though-working which says that firmware is only needed for BCM5705 TSO, BCM5703/BCM5704 TSO and BCM5701A0 (Maybe they assumed that all models need firmware to work). This is BCM5782 which continues to work even after deleting the entire /lib/firmware/tigon/ directory from a non-free distro. The stack exchange post also mentions a problem with the light indicator, which i did not encounter.

Parabola

Lack of the driver means the presence of the card doesn't appear in dmesg, only lspci works; it can't be modprobed either.

lspci -vvnnk (Linux-libre kernel)

05:02.0 Ethernet controller [0200]: Broadcom Inc. and subsidiaries NetXtreme BCM5782 Gigabit Ethernet [14e4:1696] (rev 03)
        Subsystem: Hewlett-Packard Company d530 CMT (DG746A) [103c:12bc]
        Control: I/O- Mem- BusMaster+ SpecCycle- MemWINV- VGASnoop- ParErr- Stepping- SERR+ FastB2B- DisINTx-
        Status: Cap+ 66MHz+ UDF- FastB2B+ ParErr- DEVSEL=medium >TAbort- <TAbort- <MAbort- >SERR- <PERR- INTx-
        Latency: 64 (16000ns min), Cache Line Size: 64 bytes
        Interrupt: pin A routed to IRQ 255
        Region 0: Memory at f8d00000 (64-bit, non-prefetchable) [disabled] [size=64K]
        Capabilities: [48] Power Management version 2
                Flags: PMEClk- DSI- D1- D2- AuxCurrent=0mA PME(D0-,D1-,D2-,D3hot+,D3cold+)
                Status: D0 NoSoftRst- PME-Enable- DSel=0 DScale=1 PME-
        Capabilities: [50] Vital Product Data
                Product Name: Broadcom NetXtreme Gigabit Ethernet Controller
                Read-only fields:
                        [PN] Part number: BCM95782A50
                        [EC] Engineering changes: 106679-15
                        [SN] Serial number: 0123456789
                        [MN] Manufacture ID: 14e4
                        [RV] Reserved: checksum bad, 25 byte(s) reserved
                Read/write fields:
                        [YA] Asset tag: XYZ01234567
                        [RW] Read-write area: 107 byte(s) free
                End
        Capabilities: [58] MSI: Enable- Count=1/8 Maskable- 64bit+
                Address: dfff6fffeffffefc  Data: d57f

Devuan (non-free repos disabled)

For some reason unlike other cards it fails to configure the interface during the boot so the command must be used afterwards. This is the log produced in dmesg by

dhclient

[  245.248939] tg3 0000:05:02.0: firmware: failed to load tigon/tg3_tso5.bin (-2)
[  245.248950] tg3 0000:05:02.0: Direct firmware load for tigon/tg3_tso5.bin failed with error -2
[  245.248957] tg3 0000:05:02.0 eth1: TSO capability disabled
[  245.393488] IPv6: ADDRCONF(NETDEV_UP): eth1: link is not ready
[  248.442986] tg3 0000:05:02.0 eth1: Link is up at 1000 Mbps, full duplex
[  248.443002] tg3 0000:05:02.0 eth1: Flow control is on for TX and on for RX
[  248.443112] IPv6: ADDRCONF(NETDEV_CHANGE): eth1: link becomes ready
[  249.565268] tg3 0000:05:02.0 eth1: Link is down
[  252.462527] tg3 0000:05:02.0 eth1: Link is up at 1000 Mbps, full duplex
[  252.462542] tg3 0000:05:02.0 eth1: Flow control is on for TX and on for RX
[  253.591737] tg3 0000:05:02.0 eth1: Link is down
[  256.572853] tg3 0000:05:02.0 eth1: Link is up at 1000 Mbps, full duplex
[  256.572868] tg3 0000:05:02.0 eth1: Flow control is on for TX and on for RX
[  257.693698] tg3 0000:05:02.0 eth1: Link is down
[  260.065877] tg3 0000:05:02.0 eth1: Link is up at 100 Mbps, full duplex
[  260.065891] tg3 0000:05:02.0 eth1: Flow control is on for TX and on for RX

The lease is longer than usual (5-7 seconds) but after that it works perfectly and no proprietary software is used!