I switched between a couple of strategies while hunting for both parts and cards, you may have already tried these.
Coming across the parts is naturally going to happen as you progress in the other achievements, but what's more important is that you beat the level you're on after you find a new part. Otherwise you'll see the part floating out of your debris, mocking you. I recommend that before you start to focus on hunting for parts, get your ship fully upgraded in the hangar. Then, play through the highest level that you can consistently beat on the hardest difficulty you have unlocked. To save time and risk, Level 7 on Insane was the best option for me because I could use the Ace of Spades without worrying about a boss or particularly troublesome enemies. You also get to farm stars while you work, because unlike parts and power-ups you get to keep the stars even if you die (it's because you get to take them to Heaven with you).
There are some obvious and not-so-obvious ways to increase your chances of finding parts (and cards). You might want delay part hunting until you've collected Card #12, Increased Ship Part Drop Rate. Of course this depends on the notoriously stingy RNG. I unlocked the Ace of Spades as my 3rd or 4th ship I think. While the parts you collect are random, I think certain ships are more likely to appear. The Ace of Spades is a great ship to make you better at the game because it dies easily, and while you are trying to complete all the levels with it you will pretty much always get Untouched as a bonus. Luc If you hate the Ace or don't have it, complete enough Objectives (you can track these from the main menu) to unlock Lucy Clover to increase your odds. She also increases the likely of a Lucky Shield (Card #3) proc as a bonus. I'm not sure if combining her with the Ace adds to your luck or if they overlap. Since it's more important to survive the level anyway, I usually used Burton Panic with the Ace; his shields and megabombs saved me 4,543,632 times.
On lower levels at lower difficulties you will almost always get a timed card instead of a part or permanent card. Increased Card Drop Rate also seems to work for parts, but I didn't math it out or anything. More importantly, once you have a timed card, anything else that drops has to be a permanent card or aircraft part. I went after higher and harder levels during these times, and splurged on power-ups to help survive.