You pick a single race regardless of the heritage you want for your character.
The PHB and other sources describe a finite set of races. They are not meant to be mixed or combined piecemeal. If one of a character's parents are an Elf or an Orc, you should pick either Half-Elf or Half-Orc. For racial mixes where there are no Half-breeds described, you'll have to pick one or the other. And to echo Grumbleputty, you are free to describe your character with racial features from other races so long as they are within the setting (DM approval) and don't provide mechanical advantages (speed, weapon training, etc.).
Note that there are minimal rules in the DMG for creating or modifying races, but your DM will need to be very careful to keep everything balanced. At your DM's discretion, you might be able to combine 2 races in a fair way.
(These are examples, not actual rules.)
For instance, if you want to combine Elf with Dwarf and start off with Half-Elf, you should be giving up something significant from Half-Elf in order to get something significant from Dwarf. Fortunately, it's pretty easy to remove Skill Versatility to get a Dwarf feature since that is a substantial mechanical sacrifice which doesn't hurt the Elf flavor much. It might be fair to inherit Dwarven Resilience in exchange (but once again, completely up to the DM). For Dragonborn, you could lose your Breath Weapon in order to inherit Darkvision. Be careful not to get too much for what you give up though. Your DM would also have to decide which races you qualify for if you are allowed to use the optional racial feats from XGtE.