Judging by the photo it does not look like it will fit the HP slimline 410-010, Will it fit?

I can't answer for a specific model, but I did order and install this in an HP Slim case. It is a noisy power supply but I haven't had any problems with it.

Are most of the parts new? or refurbished?

Hello, Thank you for contacting us with your question. The item is brand new. It will come closed in the original retail package with manufacturer seal. Normally, if the item was refurbished or open box it would be listed on the product page. Thank you, Newegg Support

Questions & Answers: Slimline Power Supply Upgrade for SFF Desktop Computer - Fits: HP Pavilion S5000, S5100BR, S5100LA, S5100Z CTO, S510

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Slimline Power Supply Upgrade for SFF Desktop Computer - Fits: HP Pavilion S5000, S5100BR, S5100LA, S5100Z CTO, S510

This PSU only fits SFF desktop PCs.
1 x Main connector (20+4Pin) This works with both 20 pin and 24 pin main connectors!
1 x 12V P4 connector, 3 x 4-pin Molex Connector, 2 x SATA(Serial ATA) connector, 1 x Floppy connector
300 Watts - This is an upgrade with additional wattage for additional devices, this replaces lower wattage power supplies

$79^.99 –