Carrots and parsnips are both root crops and should be planted directly into the soil, not transplanted. They are often sown early in the season but often do better by being sown later in April or early May.

Carrots: sow 1cm (½in) deep in rows 15cm-30cm (6-12in) apart. By sowing thinly you can avoid thinning out. Aim for plants 5-7.5cm (2-3in) apart. Thin if needed at the seedling stage.  It is difficult not to sow too thickly!

Parsnips: sow thinly or sow three seeds at 15cm (6in) intervals, 13mm (0.5in) deep in rows 30cm (12in) apart.  When the seedlings are about 2.5cm (1in) high thin out leaving one seedling 15cm (6in) apart.

Take a look at the seedbed article to prepare an area for sowing carrots and parsnips ...
Seedbeds - Preparation for Kale, Calabrese, Leeks and root Crops

Parsnips are in the ground for the full season but carrots can be sown at intervals.